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Calculus Volume 3

Review Exercises

Calculus Volume 3Review Exercises
  1. Preface
  2. 1 Parametric Equations and Polar Coordinates
    1. Introduction
    2. 1.1 Parametric Equations
    3. 1.2 Calculus of Parametric Curves
    4. 1.3 Polar Coordinates
    5. 1.4 Area and Arc Length in Polar Coordinates
    6. 1.5 Conic Sections
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  3. 2 Vectors in Space
    1. Introduction
    2. 2.1 Vectors in the Plane
    3. 2.2 Vectors in Three Dimensions
    4. 2.3 The Dot Product
    5. 2.4 The Cross Product
    6. 2.5 Equations of Lines and Planes in Space
    7. 2.6 Quadric Surfaces
    8. 2.7 Cylindrical and Spherical Coordinates
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  4. 3 Vector-Valued Functions
    1. Introduction
    2. 3.1 Vector-Valued Functions and Space Curves
    3. 3.2 Calculus of Vector-Valued Functions
    4. 3.3 Arc Length and Curvature
    5. 3.4 Motion in Space
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  5. 4 Differentiation of Functions of Several Variables
    1. Introduction
    2. 4.1 Functions of Several Variables
    3. 4.2 Limits and Continuity
    4. 4.3 Partial Derivatives
    5. 4.4 Tangent Planes and Linear Approximations
    6. 4.5 The Chain Rule
    7. 4.6 Directional Derivatives and the Gradient
    8. 4.7 Maxima/Minima Problems
    9. 4.8 Lagrange Multipliers
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  6. 5 Multiple Integration
    1. Introduction
    2. 5.1 Double Integrals over Rectangular Regions
    3. 5.2 Double Integrals over General Regions
    4. 5.3 Double Integrals in Polar Coordinates
    5. 5.4 Triple Integrals
    6. 5.5 Triple Integrals in Cylindrical and Spherical Coordinates
    7. 5.6 Calculating Centers of Mass and Moments of Inertia
    8. 5.7 Change of Variables in Multiple Integrals
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  7. 6 Vector Calculus
    1. Introduction
    2. 6.1 Vector Fields
    3. 6.2 Line Integrals
    4. 6.3 Conservative Vector Fields
    5. 6.4 Green’s Theorem
    6. 6.5 Divergence and Curl
    7. 6.6 Surface Integrals
    8. 6.7 Stokes’ Theorem
    9. 6.8 The Divergence Theorem
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  8. 7 Second-Order Differential Equations
    1. Introduction
    2. 7.1 Second-Order Linear Equations
    3. 7.2 Nonhomogeneous Linear Equations
    4. 7.3 Applications
    5. 7.4 Series Solutions of Differential Equations
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  9. A | Table of Integrals
  10. B | Table of Derivatives
  11. C | Review of Pre-Calculus
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
  13. Index

Review Exercises

True or False? Justify your answer with a proof or a counterexample.

416 .

a b c d f ( x , y ) d y d x = c d a b f ( x , y ) d y d x a b c d f ( x , y ) d y d x = c d a b f ( x , y ) d y d x

417 .

Fubini’s theorem can be extended to three dimensions, as long as ff is continuous in all variables.

418 .

The integral 02π01r1dzdrdθ02π01r1dzdrdθ represents the volume of a right cone.

419 .

The Jacobian of the transformation for x=u22v,y=3v2uvx=u22v,y=3v2uv is given by −4u2+6u+4v.−4u2+6u+4v.

Evaluate the following integrals.

420 .

R ( 5 x 3 y 2 y 2 ) d A , R = { ( x , y ) | 0 x 2 , 1 y 4 } R ( 5 x 3 y 2 y 2 ) d A , R = { ( x , y ) | 0 x 2 , 1 y 4 }

421 .

D y 3 x 2 + 1 d A , D = { ( x , y ) | 0 x 1 , x y x } D y 3 x 2 + 1 d A , D = { ( x , y ) | 0 x 1 , x y x }

422 .

Dsin(x2+y2)dADsin(x2+y2)dA where DD is a disk of radius 22 centered at the origin

423 .

0 1 y 1 x y e x 2 d x d y 0 1 y 1 x y e x 2 d x d y

424 .

−1 1 0 z 0 x z 6 d y d x d z −1 1 0 z 0 x z 6 d y d x d z

425 .

R3ydV,R3ydV, where R={(x,y,z)|0x1,0yx,0z9y2}R={(x,y,z)|0x1,0yx,0z9y2}

426 .

0 2 0 2 π r 1 r d z d θ d r 0 2 0 2 π r 1 r d z d θ d r

427 .

0 2 π 0 π / 2 1 3 ρ 2 sin ( φ ) d ρ d φ d θ 0 2 π 0 π / 2 1 3 ρ 2 sin ( φ ) d ρ d φ d θ

428 .

0 1 1 x 2 1 x 2 1 x 2 y 2 1 x 2 y 2 d z d y d x 0 1 1 x 2 1 x 2 1 x 2 y 2 1 x 2 y 2 d z d y d x

For the following problems, find the specified area or volume.

429 .

The area of region enclosed by one petal of r=cos(4θ).r=cos(4θ).

430 .

The volume of the solid that lies between the paraboloid z=2x2+2y2z=2x2+2y2 and the plane z=8.z=8.

431 .

The volume of the solid bounded by the cylinder x2+y2=16x2+y2=16 and from z=1z=1 to z+x=2.z+x=2.

432 .

The volume of the intersection between two spheres of radius 1, the top whose center is (0,0,0.25)(0,0,0.25) and the bottom, which is centered at (0,0,0).(0,0,0).

For the following problems, find the center of mass of the region.

433 .

ρ(x,y)=xyρ(x,y)=xy on the circle with radius 11 in the first quadrant only.

434 .

ρ(x,y)=(y+1)xρ(x,y)=(y+1)x in the region bounded by y=ex,y=ex, y=0,y=0, and x=1.x=1.

435 .

ρ(x,y,z)=zρ(x,y,z)=z on the inverted cone with radius 22 and height 2.2.

436 .

The volume an ice cream cone that is given by the solid above z=(x2+y2)z=(x2+y2) and below z2+x2+y2=z.z2+x2+y2=z.

The following problems examine Mount Holly in the state of Michigan. Mount Holly is a landfill that was converted into a ski resort. The shape of Mount Holly can be approximated by a right circular cone of height 11001100 ft and radius 60006000 ft.

437 .

If the compacted trash used to build Mount Holly on average has a density 400lb/ft3,400lb/ft3, find the amount of work required to build the mountain.

438 .

In reality, it is very likely that the trash at the bottom of Mount Holly has become more compacted with all the weight of the above trash. Consider a density function with respect to height: the density at the top of the mountain is still density 400lb/ft3400lb/ft3 and the density increases. Every 100100 feet deeper, the density doubles. What is the total weight of Mount Holly?

The following problems consider the temperature and density of Earth’s layers.

439 .

[T] The temperature of Earth’s layers is exhibited in the table below. Use your calculator to fit a polynomial of degree 33 to the temperature along the radius of the Earth. Then find the average temperature of Earth. (Hint: begin at 00 in the inner core and increase outward toward the surface)

Source: http://www.enchantedlearning.com/subjects/astronomy/planets/earth/Inside.shtml
Layer Depth from center (km) Temperature °C°C
Rocky Crust 0 to 40 0
Upper Mantle 40 to 150 870
Mantle 400 to 650 870
Inner Mantel 650 to 2700 870
Molten Outer Core 2890 to 5150 4300
Inner Core 5150 to 6378 7200
440 .

[T] The density of Earth’s layers is displayed in the table below. Using your calculator or a computer program, find the best-fit quadratic equation to the density. Using this equation, find the total mass of Earth.

Source: http://hyperphysics.phy-astr.gsu.edu/hbase/geophys/earthstruct.html
Layer Depth from center (km) Density (g/cm3)
Inner Core 00 12.9512.95
Outer Core 12281228 11.0511.05
Mantle 34883488 5.005.00
Upper Mantle 63386338 3.903.90
Crust 63786378 2.552.55

The following problems concern the Theorem of Pappus (see Moments and Centers of Mass for a refresher), a method for calculating volume using centroids. Assuming a region R,R, when you revolve around the x-axisx-axis the volume is given by Vx=2πAy,Vx=2πAy, and when you revolve around the y-axisy-axis the volume is given by Vy=2πAx,Vy=2πAx, where AA is the area of R.R. Consider the region bounded by x2+y2=1x2+y2=1 and above y=x+1.y=x+1.

441 .

Find the volume when you revolve the region around the x-axis.x-axis.

442 .

Find the volume when you revolve the region around the y-axis.y-axis.

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