Worksheet 15.2 – 15.3

Double Integrals, Triple Integrals

Math 251

Ioanna Mavrea

Note: Although some formulas from sections 15.2 and 15.3 are shown below, no formulas will be

provided in quizzes and exams. Students are responsible for all material contained in these sections,

whether that is included here or not.

Double Integrals

●

Area of R = ∬ dA

●

Average value f¯ =

●

Average value f¯ =

R

1

f (x, y) dA

Area of R ∬R

Triple Integrals

●

Volume of D = ∭ dV

D

1

f (x, y, z) dV

Volume of D ∭D

PART A

1.

Let R be the region bounded by the curves y = x2 , y = 12 − x, and

y = 4x + 12 (shaded region on the left). Set up and evaluate a double

integral that calculates the area of region R.

2. Consider the solid S below the surface f (x, y) = 2 + y1 and above the region R in the xy-plane

bounded by the lines y = x, y = 8 − x, and y = 1. Find the volume of the solid two ways:

a) Set up and evaluate a double integral, and b) Set up and evaluate a triple integral. (Notice

that f > 0 on R.)

3. Consider the double integral

8

2

∫0 ∫ √

3 y

a) Sketch the region of integration.

√

x4 + 1 dxdy.

b) Evaluate the integral.

4. Draw the region of integration and write the following integral as a single iterated integral.

1

0

e

e

a) ∫ ∫ f (x, y) dxdy + ∫ ∫ f (x, y) dxdy b) ∫ ∫

0

ey

−1 e−y

−4 0

0

√

16−x2

f (x, y) dydx+∫

4

0

5. Evaluate the integrals.

2

4

a) ∫ ∫ e

0

x2

x

√

y

dydx

b) ∫

0

√

3 π

∫y

√

3 π

x4 cos(x2 y) dxdy

∫0

4−x

f (x, y) dydx

Worksheet 15.2 – 15.3

Double Integrals, Triple Integrals

Math 251

Ioanna Mavrea

PART B

1. Use a triple integral to find the volume of the following solids.

a) B is the solid in the first octant bounded by the plane 2x + 3y + 6z = 12 and the coordinate

planes.

b) D is the solid in the first octant bounded by y 2 + z 2 = 1 and y = x.

2. Let B be the solid in the first octant bounded by x2 + z 2 = 4, x + y = 2, and 2x + y = 6.

a) Sketch the solid B

b) Compute the integral ∭ z dV .

B

3. Find the volume of sphere of radius R in three ways.

a) Set up and evaluate a single integral.

b) Set up and evaluate a double integral.

c) Set up and evaluate a triple integral.

4. Evaluate the integrals.

2

1

1

2

4

2

a) ∫ ∫ ∫ sinh(z 2 ) dzdydx

y

0

0

b) ∫ ∫ ∫ yzex dxdydz

z

0

0

3

5. In each case, a) sketch the solid B and b) evaluate the integral ∭ dV .

B

a) Let B be the solid in the first octant bounded by the graph of x + y + z = 1 and x + y + 2z = 1.

b) Let B be the region between the paraboloids z = x2 + y 2 and z = 4 − x2 − y 2 .

6. Let B be the region below the hemisphere of radius 3 and above the triangle in the xy-plane

bounded by x = 1, y = 0, and y = x. Evaluate the integral

∭B f (x, y, z) dV

where f (x, y, z) = z.

2

Worksheet 12.7

Cylindrical and Spherical Coordinates

Math 251

Ioanna Mavrea

Cylindrical Coordinates (r, θ, z)

Conversions from Cylindrical to Cartesian (rectangular)

In the cylindrical coordinate system, a point P (x, y, z) in IR3 is represented by the ordered triple (r, θ, z), where

x = r cos θ,

y = r sin θ,

z = z.

y

So, we have x2 + y 2 = r2 , and tan θ = . We also have the restriction

x

that 0 ≤ θ ≤ 2π.

Spherical Coordinates (ρ, θ, φ)

Conversions from Spherical to Cartesian (rectangular)

In the spherical coordinate system, a point P (x, y, z) in IR3 is represented by the ordered triple (ρ, θ, φ), where

x = ρ sin φ cos θ,

y = ρ sin φ sin θ,

z = ρ cos φ

Notice that x2 + y 2 + z 2 = ρ2 and r = ρ sin φ. In addition, since ρ is the

distance from the point P to the origin, and φ is the angle down from

the positive z-axis, we have the following restrictions:

0 ≤ φ ≤ π and ρ ≥ 0.

PART C

1. Describe the surface given by the cylindrical equation a) r = k, and b) θ = k (k is a constant).

2. Describe the surface given by the spherical equation a) ρ = ρ0 , and b) φ = φ0 , where ρ0 and φ0

are arbitrary constants.

3. Identify the surface by converting the given equation into a rectangular equation.

a) r = 3 cos θ

b) ρ = 3 cos φ

c) r2 − 2z 2 = 1

4. Convert into a (i) cylindrical equation and (ii) spherical equation. Simplify your answers.

a) z = 2×2 + 2y 2

b) x2 + y 2 + 2z 2 = 3

5. Write in cylindrical and spherical coordinates the inequalitites for the solid region

{(x, y, z) ∶ x2 + y 2 + z 2 ≤ 1, x ≥ 0, y ≥ 0, z ≥ 0}

6. Show that the cylindrical equation r2 (1 − 2 sin2 θ) + z 2 = 1 is a hyperboloid of one sheet.

7. How does the surface with spherical equation ρ2 (1 + A cos2 φ) = 1 depend on the constant A?

3

Purchase answer to see full

attachment

Double Integrals, Triple Integrals

Math 251

Ioanna Mavrea

Note: Although some formulas from sections 15.2 and 15.3 are shown below, no formulas will be

provided in quizzes and exams. Students are responsible for all material contained in these sections,

whether that is included here or not.

Double Integrals

●

Area of R = ∬ dA

●

Average value f¯ =

●

Average value f¯ =

R

1

f (x, y) dA

Area of R ∬R

Triple Integrals

●

Volume of D = ∭ dV

D

1

f (x, y, z) dV

Volume of D ∭D

PART A

1.

Let R be the region bounded by the curves y = x2 , y = 12 − x, and

y = 4x + 12 (shaded region on the left). Set up and evaluate a double

integral that calculates the area of region R.

2. Consider the solid S below the surface f (x, y) = 2 + y1 and above the region R in the xy-plane

bounded by the lines y = x, y = 8 − x, and y = 1. Find the volume of the solid two ways:

a) Set up and evaluate a double integral, and b) Set up and evaluate a triple integral. (Notice

that f > 0 on R.)

3. Consider the double integral

8

2

∫0 ∫ √

3 y

a) Sketch the region of integration.

√

x4 + 1 dxdy.

b) Evaluate the integral.

4. Draw the region of integration and write the following integral as a single iterated integral.

1

0

e

e

a) ∫ ∫ f (x, y) dxdy + ∫ ∫ f (x, y) dxdy b) ∫ ∫

0

ey

−1 e−y

−4 0

0

√

16−x2

f (x, y) dydx+∫

4

0

5. Evaluate the integrals.

2

4

a) ∫ ∫ e

0

x2

x

√

y

dydx

b) ∫

0

√

3 π

∫y

√

3 π

x4 cos(x2 y) dxdy

∫0

4−x

f (x, y) dydx

Worksheet 15.2 – 15.3

Double Integrals, Triple Integrals

Math 251

Ioanna Mavrea

PART B

1. Use a triple integral to find the volume of the following solids.

a) B is the solid in the first octant bounded by the plane 2x + 3y + 6z = 12 and the coordinate

planes.

b) D is the solid in the first octant bounded by y 2 + z 2 = 1 and y = x.

2. Let B be the solid in the first octant bounded by x2 + z 2 = 4, x + y = 2, and 2x + y = 6.

a) Sketch the solid B

b) Compute the integral ∭ z dV .

B

3. Find the volume of sphere of radius R in three ways.

a) Set up and evaluate a single integral.

b) Set up and evaluate a double integral.

c) Set up and evaluate a triple integral.

4. Evaluate the integrals.

2

1

1

2

4

2

a) ∫ ∫ ∫ sinh(z 2 ) dzdydx

y

0

0

b) ∫ ∫ ∫ yzex dxdydz

z

0

0

3

5. In each case, a) sketch the solid B and b) evaluate the integral ∭ dV .

B

a) Let B be the solid in the first octant bounded by the graph of x + y + z = 1 and x + y + 2z = 1.

b) Let B be the region between the paraboloids z = x2 + y 2 and z = 4 − x2 − y 2 .

6. Let B be the region below the hemisphere of radius 3 and above the triangle in the xy-plane

bounded by x = 1, y = 0, and y = x. Evaluate the integral

∭B f (x, y, z) dV

where f (x, y, z) = z.

2

Worksheet 12.7

Cylindrical and Spherical Coordinates

Math 251

Ioanna Mavrea

Cylindrical Coordinates (r, θ, z)

Conversions from Cylindrical to Cartesian (rectangular)

In the cylindrical coordinate system, a point P (x, y, z) in IR3 is represented by the ordered triple (r, θ, z), where

x = r cos θ,

y = r sin θ,

z = z.

y

So, we have x2 + y 2 = r2 , and tan θ = . We also have the restriction

x

that 0 ≤ θ ≤ 2π.

Spherical Coordinates (ρ, θ, φ)

Conversions from Spherical to Cartesian (rectangular)

In the spherical coordinate system, a point P (x, y, z) in IR3 is represented by the ordered triple (ρ, θ, φ), where

x = ρ sin φ cos θ,

y = ρ sin φ sin θ,

z = ρ cos φ

Notice that x2 + y 2 + z 2 = ρ2 and r = ρ sin φ. In addition, since ρ is the

distance from the point P to the origin, and φ is the angle down from

the positive z-axis, we have the following restrictions:

0 ≤ φ ≤ π and ρ ≥ 0.

PART C

1. Describe the surface given by the cylindrical equation a) r = k, and b) θ = k (k is a constant).

2. Describe the surface given by the spherical equation a) ρ = ρ0 , and b) φ = φ0 , where ρ0 and φ0

are arbitrary constants.

3. Identify the surface by converting the given equation into a rectangular equation.

a) r = 3 cos θ

b) ρ = 3 cos φ

c) r2 − 2z 2 = 1

4. Convert into a (i) cylindrical equation and (ii) spherical equation. Simplify your answers.

a) z = 2×2 + 2y 2

b) x2 + y 2 + 2z 2 = 3

5. Write in cylindrical and spherical coordinates the inequalitites for the solid region

{(x, y, z) ∶ x2 + y 2 + z 2 ≤ 1, x ≥ 0, y ≥ 0, z ≥ 0}

6. Show that the cylindrical equation r2 (1 − 2 sin2 θ) + z 2 = 1 is a hyperboloid of one sheet.

7. How does the surface with spherical equation ρ2 (1 + A cos2 φ) = 1 depend on the constant A?

3

Purchase answer to see full

attachment

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