Find the sine of the angle between t⃗ and u⃗ .

MA 243 Calculus III Fall 2015 Dr. E. Jacobs Assignments

These assignments are keyed to Edition 7E of James Stewart’s “Calculus” (Early Transcendentals)

Assignment 1. Spheres and Other Surfaces

Read 12.1 - 12.2 and 12.6 You should be able to do the following problems: Section 12.1/Problems 11 - 18, 20 - 22 Section 12.6/Problems 1 - 48 Hand in the following problems:

1. The following equation describes a sphere. Find the radius and the coordinates of the center.

x2 + y2 + z2 = 2(x+ y + z) + 1

2. A particular sphere with center (−3, 2, 2) is tangent to both the xy-plane and the xz-plane. It intersects the xy-plane at the point (−3, 2, 0). Find the equation of this sphere. 3. Suppose (0, 0, 0) and (0, 0,−4) are the endpoints of the diameter of a sphere. Find the equation of this sphere.

4. Find the equation of the sphere centered around (0, 0, 4) if the sphere passes through the origin.

5. Describe the graph of the given equation in geometric terms, using plain, clear language:

z = √ 1− x2 − y2

Sketch each of the following surfaces

6. z = 2− 2 √ x2 + y2

7. z = 1− y2

8. z = 4− x− y

9. z = 4− x2 − y2

10. x2 + z2 = 16

Assignment 2. Dot and Cross Products

Read 12.3 and 12.4 You should be able to do the following problems: Section 12.3/Problems 1 - 28 Section 12.4/Problems 1 - 32 Hand in the following problems:

1. Let u⃗ = ⟨ 0, 12 ,

√ 3 2

⟩ and v⃗ =

⟨√ 2,

√ 3 2 ,

1 2

⟩ a) Find the dot product b) Find the cross product

2. Let u⃗ = j⃗+ k⃗ and v⃗ = i⃗+ √ 2 j⃗.

a) Calculate the length of the projection of v⃗ in the u⃗ direction. b) Calculate the cosine of the angle between u⃗ and v⃗

3. Consider the parallelogram with the following vertices:

(0, 0, 0) (0, 1, 1) (1, 0, 2) (1, 1, 3)

a) Find a vector perpendicular to this parallelogram. b) Use vector methods to find the area of this parallelogram.

4. Use the dot product to find the cosine of the angle between the diagonal of a cube and one of its edges.

5. Let L be the line that passes through the points (0, − √ 3 , −1) and (0,

√ 3 , 1). Let θ be the

angle between L and the vector u⃗ = 1√ 2 ⟨0, 1, 1⟩. Calculate θ (to the nearest degree).

Assignment 3. Lines and Planes

Read 12.5 You should be able to do the following problems: Section 12.5/Problems 1 - 58 Hand in the following problems:

1a. Find the equation of the line that passes through (0, 0, 1) and (1, 0, 2). b. Find the equation of the plane that passes through (1, 0, 0) and is perpendicular to the line in part (a).

2. The following equation describes a straight line:

r⃗(t) = ⟨1, 1, 0⟩+ t⟨0, 2, 1⟩

a. Find the angle between the given line and the vector u⃗ = ⟨1,−1, 2⟩. b. Find the equation of the plane that passes through the point (0, 0, 4) and is perpendicular to the given line.

3. The following two lines intersect at the point (1, 4, 4)

r⃗ = ⟨1, 4, 4⟩+ t⟨0, 1, 0⟩ r⃗ = ⟨1, 4, 4⟩+ t⟨3, 5, 4⟩

a. Find the angle between the two lines.

b. Find the equation of the plane that contains every point on both lines.

4. The following equation describes a straight line:

⟨x, y, z⟩ = ⟨−1, 0, −2⟩+ t⟨1, 2, 2⟩

Find the coordinates of the point where this line intersects the y-axis.

5. There is a plane that contains the y-axis as well as every point on the line described in problem 4. Find the equation of this plane.

Assignment 4. Vector Functions and Space Curves

Read 13.1 - 13.4 You should be able to do the following problems: Section 13.1/Problems 7 - 30 Section 13.2/Problems 1 - 28, 31 Section 13.3/Problems 1 - 6 Section 13.4/Problems 3 - 16, 36 - 42 Hand in the following problems:

1. Suppose the position of a particle after t seconds is given by the following vector equation: r⃗(t) = ⟨1 + cos 2πt, sin 2πt⟩ At t = 14 sec., compute each of the following vectors:

a) The velocity vector b) The acceleration vector c) The unit tangent vector

d) calculate the arc length for 0 ≤ t ≤ 14 2. Consider the helical path described by the following vector equation:

⟨x, y, z⟩ = ⟨ cos

t

2 , sin

t

2 , t

2

⟩ At t = π, the helix passes through the point (0, 1, π/2).

a. Find a vector v⃗ that is tangent to the helix at (0, 1, π/2) b. Find the equation of the plane that passes through (0, 1, π/2) and is perpendicular to v⃗.

3. Suppose a particle is moving along a three dimensional path and that its coordinates after t seconds are given by the parametric equations:

x = e2t y = 2et z = t

Find the length of the curve between t = 0 and t = 1.

4. Let r⃗(t) be the position of a particle at time t. Suppose r⃗(t) is given by the formula:

r⃗(t) = ⟨π, cos(πt), sin(πt)⟩

a) Find the velocity vector v⃗ and the acceleration vector a⃗

b) Find the length of the projection of the acceleration vector on the velocity vector.

c) Find the distance traveled by the particle for 0 ≤ t ≤ 2.

5. When two objects travel through space along two different curves, it is often important to know if they will collide. Suppose the trajectories of the two objects are given by the vector functions:

r⃗1(t) = ⟨−2 + 3t, −4 + 4t, 2 + t⟩ r⃗2(t) = ⟨ 2t, 2t,

8

t

⟩ Will the two objects ever collide? If so, for what value of t will this happen?

Assignment 5. Partial Derivatives

Read 14.1 to 14.3 You should be able to do the following problems: Section 14.3/Problems 15 - 56, 63 - 69, 75 - 78, 80 - 81 Hand in the following problems:

For problems 1 - 5, calculate the partial derivatives ∂z∂x and ∂z ∂y

1. z = ln ( ye2x

) 2. z = 2

√ x+ x tan(2y)

3. z = y44x

4. z = x+ y

x

5. z = arctan

( x

y

) 6. Let z = 1x2y . Calculate the partial derivative

∂2z ∂x∂y

7. Let f(x, y) = e2πx sin 2πy. Calculate each of the following partial derivative expressions:

∂2f

∂x∂y

∂2f

∂x2 +

∂2f

∂y2

8. If f(x, y, z) = xy − yz3

6 calculate the third partial derivatives ∂3f

∂z2∂y and ∂3f

∂x∂y∂z

9. The electric field E is a function of charge q and distance x as described by the following formula:

E = kq

x2 (where k is a constant)

Calculate the following second partial derivatives:

∂2E

∂x2 ∂2E

∂x∂q

∂2E

∂q2

10. If m is a person’s mass (in kilograms) and h is this person’s height (in cm) then the body- mass-index B is given by:

B = m

h2

Calculate the following partial derivatives:

∂B

∂m

∂B

∂h

∂2B

∂m∂h

∂2B

∂m2

Assignment 6. Differential, Tangent Planes

Read 14.4 You should be able to do the following problems: Section 14.4/Problems 1 - 19, 25 - 41 Hand in the following problems:

Several of the problems below refer to the equation of the tangent plane. As shown in class, the equation of the plane that is tangent to the surface z = f(x, y) at the point (x0, y0) is given by:

z = f(x0, y0) + a(x− x0) + b(y − y0) where a = ∂f

∂x (x0, y0) and b =

∂f

∂y (x0, y0)

1. Find the equation of the plane that is tangent to the surface 12x 2 + 12y

2 + z = 18 at the point (2, 4, 8).

2. Find the equation of the plane tangent to the surface z = 1 + e−x + y2 at the point (0, 1, 3).

3. Suppose z is a function of x and y and is given by:

z = x2ey

Suppose (x, y) changes from (2, 1) to (2.50, 1.25). Calculate the differential dz

4. The Ideal Gas Law relating pressure P , volume V and temperature T is P = nRTV . For simplicity, assume that the constant nR is equal to 1 (newton-meters/degree). Calculate the differential dP if V changes from 0.25 to 0.255 cubic meters and T changes from 4 to 4.03 degrees.

5. The kinetic energy E of a moving object is a function of the mass m and the velocity V and is given by the formula:

E = 1

2 mv2

Suppose we start with a mass of 10 kilograms going at a velocity of 10 meters/second. If the mass decreases to 9 kilograms and the velocity increases to 11 meters/second, find the value of the differential dE (in kg m2/sec2).

Assignment 7. The Chain Rule for Partial Derivatives

Read 14.5 You should be able to do the following problems: Section 14.5/Problems 1 - 13, 17 - 26, 35 - 44 Hand in the following problems:

1. Suppose two resistors are connected in parallel and that the first resistor is x ohms and the second resistor is y ohms. The total resistance is given by:

R = xy

x+ y

If the resistance of the first resistor is increasing at 1 ohm/minute and the second resistor is increasing at 2 ohm/minute, use the Chain Rule for Partial Derivatives to find dRdt at the point in time when x and y are both 1 ohm.

2. The pressure P , temperature T and volume V of a gas in a container is given by:

P = kT

V where k is a constant

Temperature is measured in degrees (deg), volume is measured in liters (ℓ) and pressure is measured in kilopascals (kPa). For simplicity, we will assume that k = 1 kPa-ℓ/deg. Suppose the temperature is increasing at 3 deg/hour and the volume is increasing at 4 ℓ/hour. Use the Chain Rule for Partial Derivatives to calculate dPdt (the derivative of pressure with respect to time) at the point in time when the volume is 10 ℓ and the temperature is 20 deg.

3. A rectangular block of ice is x cm by x cm at the base and y cm tall. The surface area is:

A = 2x2 + 4xy

Suppose this block of ice is melting with dxdt = − 1 2 cm/min and

dy dt = −

1 4 cm/min. Use the Chain

Rule for Partial Derivatives to find dAdt at the point in time when x = 4 cm and y = 6 cm.

4. Suppose z = x3y+xy3, x = r cos θ and y = r sin θ. Use the Chain Rule for Partial Derivatives to calculate ∂z∂r and

∂z ∂θ . Evaluate these derivatives at the point where r =

1 2 and θ =

π 4

5. Suppose w = f(x, y, z) where x, y and z are given by:

x = ρ cos θ sinϕ y = ρ sin θ sinϕ z = ρ cosϕ

You are not given the formula for how w depends on x, y and z, but you are given the following partial derivatives:

∂w

∂x =

y

(x+ y)2 ∂w

∂y =

−x (x+ y)2

∂w

∂z = −1

Find ∂w∂ϕ when ρ = 2, θ = π 4 and ϕ =

π 2

Assignment 8. Directional Derivatives and the Gradient

Read 14.6 You should be able to do the following problems: Section 14.6/Problems 4 - 46 Hand in the following problems:

1. For each of the following, calculate the gradient ∇f (P0) and the directional derivative Dv⃗f (P0)

a) f(x, y) = x2y, P0 = (2, 2) where v⃗ = ⟨−1, 0⟩ b) f(x, y) = 4ex sin y, P0 =

( 1, π2

) where v⃗ is a unit vector that makes an angle of π3 with the

positive x axis.

2. Captain Ralph is flying his rocket through a cloud of poisonous and corrosive gas in the atmosphere of a distant planet. The equation of the density of this gas at a point (x, y, z) is known to be:

f(x, y, z) = e−x 2−y2−4z2

a) At the point (1, 1, 0), calculate the rate that the density changes in the direction of the vector

u⃗ = 1√ 2 (⃗i+ k⃗)

b) At the point (1, 1, 0), find a vector that points in the direction that Captain Ralph should fly if he wishes to decrease the density of the poisonous gas surrounding his rocket as quickly as possible.

3. Let f(x, y) = 4− x− y2.

a. Sketch some of the level curves (contours) of this function. Indicate the direction of the gradient ∇f at several points on your sketch. b. Calculate the directional derivative Du⃗f at (2,

√ 2) in the direction of u⃗ = 13 ⟨2

√ 2, −1⟩.

c. Find a unit vector v⃗ that points in the direction that will maximize Dv⃗f(P0) at P0 = (4, 0).

4. Let f(x, y) = e1+y−x

a. Describe what the level sets of f(x, y) look like in the xy plane.

b. Let T⃗ = 1√ 2 ⟨1, 1⟩. Calculate the directional derivative DT⃗f

c. The value of the directional derivative Dv⃗f depends of which unit vector v⃗ we choose. Find the largest possible value of Dv⃗f(0, 0)

5. The equation 3x2 + 3y2 + z2 = 12 describes an ellipsoid centered around the origin. Find the equation of the plane that is tangent to this ellipsoid at the point (1, 0, 3)

Assignment 9. Max-Min Problems

Read 14.7 You should be able to do the following problems: Section 14.7/Problems 5 - 12, 39 - 55 Hand in the following problems:

Problems 1 and 2. For each of the following functions, find and classify all critical points. Sketch the surface.

1. f(x, y) = 3 + (x+ y − 3)(x− y)

2. f(x, y) = x3 − 3x 1 + y2

3. A rectangular box with no top and a volume of 6 cubic feet is to be constructed from material that costs 6 dollars per square foot for the bottom, 2 dollars per square foot for the front and back, and 1 dollar per square foot for the sides. What dimensions would minimize the cost of the box?

4. Find the dimensions of the rectangular box of maximum volume whose base is the rectangle with corners (±x,±y) in the xy-plane and whose upper corners are on the elliptic paraboloid z = 4− 9x2 − y2

5. Find the point on the surface z = y − x 2

2 that is closest to the point (4, 2, 2).

Assignment 10. Double Integrals

Read 15.1 to 15.3 You should be able to do the following problems: Section 15.2/Problems 1 - 31, Section 15.3/Problems 1 - 28 Hand in the following problems:

1. Calculate the following double integral:∫ 2 1

∫ 1 0

(x+ y)−2 dx dy

2. Calculate the following double integral:∫ 2 1

∫ e 1

1

x2y dy dx

3. Let R be the region in the xy plane bounded by y = x2 and y = 1. a. Express

∫∫ R ( 5 + 5x2

) dA as an iterated integral. Evaluate this integral.

b. Express ∫∫

R ( 5 + 5x2

) dA as an iterated integral with order reversed from the order you used

in part (a). Don’t evaluate the integral.

4. Reverse the order of integration:

∫ 2 0

∫ 1 2

√ 4−x2

− 12 √ 4−x2

f(x, y) dy dx

5. Let R be the region in the xy plane that is bounded by y = x2 and y = 4 − x2. Set up the limits of integration for

∫∫ R f(x, y) dA. Note that a formula for f(x, y) is not specified, so you

don’t have to calculate any antiderivatives.

Assignment 11. Additional Double Integral Problems

Read 15.3 You should be able to do the following problems: Section 15.3/Problems 43 - 54 Hand in the following problems:

1. Let T be the triangular region in the xy plane with vertices (0, 0), (2, 2) and (2, −2). Calculate the integral

∫∫ T ex+y dy dx

2. Let Ω be the region in the xy plane bounded by y = x, y = 2x and x = 1. Calculate the volume under the surface z = 2y1+x3 and above the region Ω.

3. The area of a region Q is given by ∫∫

Q 1 dA. Suppose that Q is the region in the xy plane for

x ≥ 0 that is inside the circle x2 + y2 = 2 and above the line y = x. Set up the iterated integrals, showing all limits of integration, for this area where we integrate:

a) first with respect to y and then with respect to x b) first with respect to x and then with respect to y

There is no need to calculate either one of these integrals. I am only interested if you get the limits of integration right.

4. Let G be the region in the xy plane that is bounded by y = 2x2−2 and y = 4−4x2 for x ≥ 0. Calculate the integral

∫∫ G 1 dA.

5. Let R be the region inside the triangle in the xy plane with vertices (0, 0), (1, −1) and (1, 1). Calculate the following double integral: ∫ ∫

R

1

1 + x2 dA

In problems 6 - 10, reverse the orders of integration

6.

∫ 1 0

∫ 2(1−x2) 2(1−x)

f(x, y) dy dx

7.

∫ ∞ 0

∫ ∞ y/2

f(x, y) dx dy

8.

∫ 2 0

∫ 1 x/2

f(x, y) dy dx

9.

∫ 3 −3

∫ 9 x2

f(x, y) dy dx

10.

∫ 1 0

∫ 1 e−x

f(x, y) dy dx

Assignment 12. Double Integrals in Polar Coordinates.

Read 15.4 You should be able to do the following problems: Section 15.4/Problems 1 - 34 Hand in the following problems:

1. Let R be the sector in the first quadrant bounded by y = 0, y = x and x2+ y2 = 4. Use polar coordinates to evaluate the double integral:∫ ∫

R

1

1 + x2 + y2 dA

2. Let R be the region in the xy-plane that is outside a circle of radius π centered around the origin.

R = {(x, y) : x2 + y2 ≥ π2 }

Calculate the following double integral by converting to polar coordinates first.∫ ∫ R

1

(π2 + x2 + y2) 2 dA

3. Use polar coordinates to evaluate the following double integral:∫ ∞ −∞

∫ ∞ 0

e−x 2−y2 dy dx

4. Let Ω be the region in the xy plane that is above the x axis and between the semicircles y =

√ 1− x2 and y =

√ 2− x2. Calculate the following double integral.∫ ∫

Ω

y

(x2 + y2) 3/2

dA

5. Convert the following double integral to polar coordinates and then calculate the integral.

∫ 1 0

∫ √2−x2 x

1 dy dx

Assignment 13. Applications of Double Integral

Read 15.5 and 15.6 You should be able to do the following problems: Section 15.5/Problems 1 - 14, Section 15.6/Problems 1 - 12

Hand in the following problems:

1. Let C be the region in the xy-plane that is inside the circle of radius 1 centered around (0, 1).

Calculate the surface area of the portion of the cone z = √ x2 + y2 that lies directly above R.

2. Let T be the region inside the triangle with vertices (0, 0, 0), (1, 0, 0) and (1, 2, 0). Use double integration to find the surface area of the portion of the plane z = 1 + x+y√

2 that is directly above

T .

3. Let Q be the region in the xy plane bounded by the lines x = 0, y = 2 and 3y− 2x = 0. Find the surface area of the portion of the surface z = x+ y

2

2 that is directly over Q.

4. Find the surface area of the portion of the cylinder y2 + z2 = 9 above the rectangle:

R = {(x, y) : 0 ≤ x ≤ 2, −3 ≤ y ≤ 3}

5. Let Ω be the sector bounded by y = x, y = −x and x = √ 9− y2. The area of this region 9π4 .

Find the coordinates of the centroid.

Assignment 14. Triple Integrals

Read 15.7 You should be able to do the following problems: Section 15.7/Problems 1 - 21, 29 - 42 Hand in the following problems:

1. Let Ω be the region in the first octant that is inside both the cylinders x2 + y2 = 1 and y2 + z2 = 1. Use a triple integral to evaluate the volume of Ω.

2. Let T be the three-dimensional region bounded by the planes z = y, z = 2− y, y = 0, x = 0 and x = 4. Calculate the volume of T

3. Reverse the order of integration by filling in the missing limits. You do not have to calculate the antiderivatives. ∫ 1

0

∫ 1−x2 0

∫ 2 0

1 dz dy dx =

∫ 2 0

∫ ? ?

∫ ? ?

1 dx dy dz

4. Let T be the three dimensional region whose volume is given by the following:

Vol(T ) =

∫ 1 0

∫ 1−z 0

∫ 2y 0

1 dx dy dz

Evaluate the volume of T by calculating the specified triple integral.

5. Let R be the triangle with vertices (−1, 0, 0), (0, 1, 0) and (0, 0, 0). Let T be the region directly above R but below the plane −2x + 2y + z = 2. The volume of T can be expressed as a triple integral

∫∫∫ T 1 dV . In each of the triple integrals below, you are given the lower limits of the

triple integral and you must fill in the correct upper limits. You are not required to calculate any antiderivatives. ∫ ∫ ∫

T

1 dV =

∫ ? −1

∫ ? 0

∫ ? 0

1 dz dy dx =

∫ ? 0

∫ ? 0

∫ ? y−1+z/2

1 dx dy dz

Assignment 15. Triple Integrals - Cylindrical and Spherical Coordinates

Read 15.8 and 15.9 You should be able to do the following problems: Section 15.8/Problems 1 - 28, Section 15.9/Problems 1 - 36, 39 - 41 Hand in the following problems:

1. Let G be the solid in the first octant bounded by the sphere x2+y2+z2 = 4 and the coordinate planes. Set up and evaluate

∫∫∫ G xyz dV using

(a) cylindrical coordinates (b) spherical coordinates

2. Let T be the three dimensional region above the xy plane, below the cone z = √

x2 + y2 and inside the sphere x2 + y2 + z2 = 1. Express the triple integral

∫∫∫ T z dV in spherical coordinates.

Do not calculate the integral.

3. Let T be the set of all points (x, y, z) that lie outside the sphere x2 + y2 + z2 = 1. Calculate the following triple integral. ∫ ∫ ∫

T

1

(x2 + y2 + z2) 2 dV

4. Evaluate the following triple integral by converting it to cylindrical coordinates and integrating.∫ 4 0

∫ 1 0

∫ √2−y2 y

dx dy dz

Note that the region of integration is inside the cylinder x2 + y2 = 2 and bounded by the planes y = x, y = 0, z = 0 and z = 4.

5. The triple integral ∫∫∫

Q

√ x2 + y2 + z2 dV given below is expressed in rectangular coordinates.

Q is a quarter of a sphere of radius 1. Calculate this integral by converting to spherical coordinates.∫ 1 −1

∫ √1−x2 0

∫ √1−x2−y2 0

√ x2 + y2 + z2 dz dy dx

6. Let S1 be the sphere of radius 1 centered around the origin. Let S2 be the sphere of radius 2 centered around the origin. Let B be the region outside S1 but inside S2. Suppose there is electric charge distributed throughout region B with charge density δ = 1x2+y2+z2 (coul/m

3). Express the total charge in region B as a triple integral and calculate it.

7. Let B be the three dimensional region below the sphere z = √ 1− x2 − y2 and above the

xy-plane. Suppose the density of the material inside B is given by: δ(x, y, z) = 8z (in kg/m3).

Let Mass(B) denote the total mass (in kilograms) of the material inside region B. Express Mass(B) as a triple integral in rectangular coordinates. Express the mass also as a triple integral in spherical coordinates. Use the spherical coordinate integral to evaluate.

8. Let G be the three dimensional region above the paraboloid z = x2 + y2 but below the plane z = 2. You are given the fact that the volume of G is 2π

Calculate the z-coordinate of the centroid of G

9. Let T be the region inside the sphere x2 + y2 + z2 = 1 for x ≥ 0, y ≥ 0 and z ≥ 0. Note that T is an eighth of a sphere. The volume of T is π6 . Find the z-coordinate of the centroid of T .

10. Calculate the volume that is above the paraboloid z = 1− x2 − y2 but below the paraboloid z = 2− 2x2 − 2y2

Assignment 16. Change of Variables in Multiple Integrals

Read 15.10 You should be able to do the following problems: Section 15.10/Problems 1 - 6, 11 - 13, 17 - 20 Hand in the following problems:

1. Find the Jacobian ∂(x,y)∂(u,v) of the following transformations:

a. x = u2 − 4v2, y = u2 + 4v2

b. x = u cos v, y = u sin v

2. Let R be the square in the xy plane with vertices (0, 0), (2, 2), (0, 4) and (−2, 2). Evaluate the integral

∫∫ R y dA using the following transformation:

x = 2u− 2v y = 2u+ 2v

3. Let R be the parallelogram with the vertices (0, 0), (2, 1), (0, 1) and (2, 2). Evaluate the integral∫∫ R (2y − x) dA using the following transformation:

x = 2u

y = u+ v

4. Let R be the triangle with vertices (0, 0), (1, 1) and (2,−1). Evaluate the integral ∫∫

R (4x+8y) dA

using the following transformation: x = 2u+ v

y = −u+ v

Assignment 17. Surface Integrals

Read 16.6 and 16.7 You should be able to do the following problems: Section 16.6/Problems 39 - 50, Section 16.7/Problems 23 - 32 Hand in the following problems:

1. The following parametric equation describes a portion of a cone:

r⃗ = ⟨z cos θ, z sin θ, z⟩ for 0 ≤ θ ≤ 2π and 0 ≤ z ≤ 1 Use the formula A =

∫∫ ∣∣ d⃗r dθ ×

d⃗r dz

∣∣ dθ dz to calculate the surface area. 2. Let Ω be the surface described by:

r⃗ = ⟨cos θ, sin θ, z⟩ for 0 ≤ θ ≤ π and 0 ≤ z ≤ 2

Let F⃗ = ⟨

x x2+y2 ,

y x2+y2 , 2z

⟩ . Evaluate the following surface integral. Show all work.∫ ∫

Ω

F⃗ • n⃗ dS

3. Let V be the tetrahedral solid with the vertices: (1, 0, 0) (0, 1, 0) (0, 0, 1) (0, 0, 0)

Let T be the triangular top of this solid. Notice that T is contained in the plane z = 1− x− y. Let F⃗ = ⟨12x, 0, −12x⟩ Calculate the surface integral: ∫ ∫

T

F⃗ • n⃗ dS

4. Let Q be the following rectangle in the xy plane :

Q = {(x, y) : 0 ≤ x ≤ 2 and 0 ≤ y ≤ π} Let Ω be the portion of the surface z = sin y that lies directly above Q. Let F⃗ = ⟨2xz, x, xy⟩ Evaluate the surface integral:∫ ∫

Ω

F⃗ • n⃗ dS

5. Let F⃗ be the vector field defined by the formula:

F⃗ = ⟨ 2x2, sinπy, 4y

⟩ Let Ω be the surface z = 1 − x2 bounded by the planes z = 0, y = 0 and y = 1. Calculate the surface integral: ∫ ∫

Ω

F⃗ • n⃗ dS

Assignment 18. Line Integrals

Read 16.1 - 16.2 You should be able to do the following problems: Section 16.2/Problems 1 - 22, 39 - 42 Hand in the following problems:

1. Consider the curve C given by the parametric equations:

x = et cos t y = et sin t z = 0 between t = 0 and t = 1.

Suppose the density at each point (x, y, z) on the curve is f(x, y, z) = 1x2+y2+z2 kilograms per

meter. The mass along curve C is ∫ C f(x, y, z) ds. Calculate this mass.

2. Let C be the section of the curve y = ex that connects (1, e) to (e, 1). Let F⃗ = ⟨x+ y, y⟩. Calculate the line integral

∫ C F⃗ • d⃗r

3. Let C1 be the path described by the equation r⃗(t) = ⟨ 2, t, 4− t2

⟩ from t = 2 to t = 0. Let F⃗

be the vector field defined by the equation F⃗ = ⟨xyz, x+ y, 3y − 2z⟩. Evaluate the following line integral. Show all work. ∫

C1

F⃗ • d⃗r

4. Let C1 and F⃗ be defined exactly as they were in problem 3. Suppose C is the closed loop that we get if we go from (2, 2, 0) to (2, 0, 4) along path C1 and then along a straight line path C2 from (2, 0, 4) to (2, 0, 0) and finally along a straight line path C3 from (2, 0, 0) to (2, 2, 0). Evaluate the following line integral. Show all work. ∮

C

F⃗ • d⃗r

5. Let F⃗ = (x2/y)⃗i+ y⃗j+ k⃗. Let C be the segment of the curve described by the following equations that passes through the points (1, 1, 0) and (e, e, 1) :

x = y, z = ln y

Evaluate ∫ C F⃗ • d⃗r

Assignment 19. Fundamental Theorem for Line Integrals

Read 16.3 You should be able to do the following problems: Section 16.3/Problems 3 - 10, 12 - 18, 21 - 22 Hand in the following problems:

1. For each vector field F⃗, find a scalar valued function ϕ = ϕ(x, y) so that F⃗ = ∇ϕ.

a) F⃗ =

⟨ y2

2x , y lnx

⟩

b) F⃗ =

⟨ y +

1

x , x

⟩

2. Let F⃗ = ⟨

y2

2x , y lnx ⟩

Calculate ∫ Γ F⃗ • d⃗r where Γ is the straight line segment connecting (1, 1, 1) to (2, 2, 3).

3. Let H be the segment of the hyperbola x2 − y2 = 1 connecting (1, 0) to (2, √ 3). Evaluate the

integral: ∫ H

( y +

1

x

) dx+ x dy

4. Let F⃗ = ⟨ 2y

√ y + 4, 3x

√ y ⟩

a) Find a scalar-valued function ϕ = ϕ(x, y) so that F⃗ = ∇ϕ b) Calculate the line integral

∫ C F⃗ • d⃗r where C is any path connecting (1, 1) to

( 1 2 , 4

) 5. Let F⃗ = ⟨exy, ex + z, y⟩. Let L be the straight line path from (1, 0, 0) to (0, π, π). Calculate the following line integral:∫

L

F⃗ • d⃗r

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