Deck 8: Applications of Integration

Question 1

The equations y = x³, y = 0, and x = 1 define the bounds of a plane region. Find the volume of the solid obtained by rotating the region about the x-axis.

A) cubic units
B) cubic units
C) cubic units
D) cubic units
E) cubic units

Answer 1

cubic units

Question 2

The equations x = 2, x = 4, y = 1/x, and y = 0 define the bounds of a region of the plane. Find the volume of the solid obtained by rotating the region about the x-axis.

A) cubic units
B) cubic units
C) cubic units
D) cubic units
E) cubic units

Answer 2

cubic units

Question 3

Find the volume of the solid obtained by rotating about the x-axis the region lying under the curve above the x-axis and to the left of the y-axis.

A) 32 $\pi$ cubic units
B) 16 $\pi$ cubic units
C) 64 $\pi$ cubic units
D) 4 $\pi$ cubic units
E) 25 $\pi$ cubic units

Answer 3

32 $\pi$ cubic units

Question 4

Find the volume of the solid obtained by rotating about the x-axis the plane region lying under the x-axis and above the curve y = x² - 2x.

A) cubic units
B) cubic units
C) cubic units
D) cubic units
E) cubic units

Question 5

The region R is bounded by y = ln x, y = 0, x = 1, and x = 2. Find the volume of the solid obtained by revolving R about the y-axis.

A) $\pi$ cubic units
B) cubic units
C) $\pi$ cubic units
D) $\pi$ cubic units
E) $\pi$ cubic units

Question 6

Find the volume of a solid generated when the region under the curve y = sin x and above the x-axis from x = 0 to x = $\pi$ is rotated about the x-axis.

A) cubic units
B) cubic units
C) cubic units
D) cubic units
E) cubic units

Question 7

Find the volume of a solid generated when the region under the curve y = sin x and above the x-axis from x = 0 to x = $\pi$ is rotated about the y-axis.

A) 2 cubic units
B) cubic units
C) 3 cubic units
D) cubic units
E) 4 cubic units

Question 8

The equations x = -1, x = 0, y = , and y = 0 define the bounds of a region of the plane. Find the volume of the solid obtained by rotating the region about the x-axis.

A) cubic units
B) cubic units
C) cubic units
D) cubic units
E) cubic units

Question 9

If R is the region enclosed by the graphs of y = f(x) and y = g(x) from x = a to x = b (as shown in the figure below), then the volume V of the solid generated by revolving the region R about the line y = -2 is V = $\pi$ dx.

Question 10

The equations y² = 4x and x² = 4y define the bounds of a plane region. Find the volume of the solid obtained by rotating the region to the right of the curve y² = 4x and above the curve x² = 4y about
(a) the x-axis and
(b) the y-axis.

A) (a) cubic units, (b) cubic units
B) (a) cubic units, (b) cubic units
C) (a) cubic units, (b) cubic units
D) (a) cubic units, (b) cubic units
E) (a) cubic units, (b) cubic units

Question 11

Find the volumes of solids generated when the ellipse + = 1 (where a > 0 and b > 0) is rotated about (a) the x-axis and (b) the y-axis.

A)
B)
C)
D)
E)

Question 12

Find the volume of the solid obtained by rotating the region inside the circle x² + y² = 6 and above the parabola y = x² about the x-axis.

A) cubic units
B) cubic units
C) cubic units
D) cubic units
E) cubic units

Question 13

Find the volume of the solid obtained by rotating the region inside the circle x² + y² = 6 and above the parabola y = x² about the y-axis.

A) 8 $\pi$ - cubic units
B) 4 $\pi$ - cubic units
C) 4 $\pi$ - cubic units
D) 8 $\pi$ - cubic units
E) 2 $\pi$ - cubic units

Question 14

The region R is the portion of the first quadrant that is below the parabola y² = 8x and above the hyperbola y² - x² = 15. Find the volume of the solid obtained by revolving R about the x-axis.

A) cubic units
B) cubic units
C) cubic units
D) cubic units
E) cubic units

Question 15

Find the volume of the solid generated by revolving the triangular region bounded by the lines y = x, y = -x, and x = a (where a > 0) about its edge x = a.

A) cubic units
B) cubic units
C) cubic units
D) cubic units
E) cubic units

Question 16

Find the volume of the solid generated when the region lying under the curve y = 4 - x² and above the x-axis is rotated about the line y = -1.

A) cubic units
B) cubic units
C) cubic units
D) cubic units
E) cubic units

Question 17

A cylindrical hole of radius r cm is drilled through the centre of a ball of radius R cm (where R > r). Find the volume of the remaining part of the ball.

A)
B)
C)
D)
E)

Question 18

Find the volume of a solid formed by revolving the disk bounded by a circle of radius a cm about a line tangent to that circle.

A) 2
B)
C) $\pi$
D) 4 $\pi$
E) 4

Question 19

The plane region bounded by the curve + = 1 is revolved about the line x = 2. Find the volume of the solid generated.

A) cubic units
B) cubic units
C) cubic units
D) cubic units
E) cubic units

Question 20

For what real values of the constant k does the region lying under the curve y = above the x-axis and to the right of the line x = 1 have infinite area but gives rise to a solid with finite volume when rotated about the x-axis?

A) < k $\le$ 1
B) < k < 1
C) < k < $\infty$
D) 0 < k <
E) 0 < k $\le$ 1

Question 21

Find the volume of the solid generated by revolving the plane region bounded by the graphs of and the line y = 3 from x = 0 to x = 2ln(2) about the x-axis.

A) 27 $\pi$ cubic units
B) $\pi$ cubic units
C) 21 $\pi$ cubic units
D) 37 $\pi$ cubic units
E) 45 $\pi$ cubic units

Question 22

If R is the region enclosed by the graphs of y = f(x) and y = g(x) from x = a to x = b (as shown in the figure below), then the volume V of the solid generated by revolving the region R about the line x = -2 is V = 2 $\pi$ dx.

Question 23

The solid generated by revolving the plane region R about the x-axis has the same volume as the solid generated by revolving the region R about the y-axis.

Question 24

Let R be the plane region enclosed by the graphs of y = f(x) and y = g(x) from x = a to x = b , where a > 0(as shown in the figure below).If the solid generated by revolving the plane region R about the x-axis has the same volume as the solid generated by revolving the region R about the y-axis , then f and g satisfy which equation for all x > 0?

A) f(x) = - g(x)
B) f(x) + g(x) = x
C) f(x) + g(x) =
D) f(x) + g(x) = 2x
E) f(x) + g(x) = $\pi$ (x -2)

Question 25

Find the volume of the solid generated by revolving the region enclosed by the graphs of y = and the x-axis from x = 0 to x = 1 about the y-axis.

A) 2 $\pi$ (2e -1)
B) 2 $\pi$
C) $\pi$ e
D)
E) $\pi$

Question 26

Find the volume of a right circular cone of base radius r and height h.

A) h cubic units
B) h cubic units
C) h cubic units
D) h cubic units
E) h cubic units

Question 27

Find the volume of an elliptical cone whose base in the horizontal xy-plane is the elliptic disk (where a > 0 and b > 0) and whose vertex is at height h directly above the centre of the base.

A) abh cubic units
B) abh cubic units
C) h cubic units
D) h cubic units
E) h cubic units

Question 28

A pyramid has a triangular base of area A and has a height of h measured perpendicular to the plane of the base. Determine the volume of the pyramid.

A) Ah cubic units
B) Ah cubic units
C) Ah cubic units
D) Ah cubic units
E) Ah $\pi$ cubic units

Question 29

A cube has edge length a cm and one corner at position O. A plane passing through the three corners of the cube that are adjacent to corner O slices the cube into two pieces. Find the volume of the smaller piece.

A)
B)
C)
D)
E)

Question 30

Find the volume of a solid whose base is the region in the first quadrant bounded by the line and the coordinate axes if every planar section perpendicular to the x-axis is a semicircle.

A) cubic units
B) cubic units
C) 4 $\pi$ cubic units
D) cubic units
E) cubic units

Question 31

The base of a certain solid is a circular disk of radius a cm. Cross-sections of the solid in planes perpendicular to a specific diameter of the base are equilateral triangles. Find the volume of the solid.

A)
B)
C) 2
D)
E) 4

Question 32

A ball of radius r has volume V(r) = $\pi$ cubic units. This volume can be regarded as a sum of volumes of concentric spherical shells having radii x units (where 0 $\le$ x $\le$ r) and thickness dx. Use this fact to find the surface area S(r) of a sphere of radius r.

A) S(r) = 4 $\pi$ square units
B) S(r) = 8 $\pi$ square units
C) S(r) = 2 $\pi$ square units
D) S(r) = 8 square units
E) S(r) = $\pi$ square units

Question 33

A certain solid S has a horizontal plane region R as its base and has height h cm measured perpendicular to R. For 0 < z < h, the volume of that part of S lying beneath the plane at height z cm above R is V(z) = 2z + z³ cm³. Find (a) the area of the cross-section of S in the plane at height z cm and (b) the area of R.

A) (a) 2 + 3z² cm², (b) 2 cm²
B) (a) 1 + 3z² cm², (b) 1 cm²
C) (a) 3 + z² cm², (b) 3 cm²
D) (a) 2 + 4z² cm², (b) 3 cm²
E) (a) 1 + 2z² cm², (b) 1 cm²

Question 34

A notch is cut out of a vertical cylindrical log of radius r cm by two planar saw cuts that meet along a horizontal line passing through the centre of the log. If the saw cuts make angles ± 30º with the horizontal (so that the angle of the notch is 60º), find the volume of wood cut out of the log in making the notch.

A) cm ³
B) cm³
C) cm³
D) cm³
E) cm³

Question 35

Let A(x) be the cross-sectional area of a solid by planes perpendicular to the x-axis. If the volume of the solid that lies between x = 1 and x = z > 1 is V = 4 + 2, find A(x).

A) 12 square units
B) + 2x - 3 square units
C) 12 square units
D) + 2x + C square units
E) 6 square units

Question 36

Find the total length of the hypocycloid + = .

A) 12a units
B) 10a units
C) 8a units
D) 6a units
E) $\pi$ a units

Question 37

Find the length of the arc y = ln between x = 1 and x = 2.

A) ln( + 2) - 1 units
B) ln( + 1) - 2 units
C) ln( - 1) - 1 units
D) ln( + 1) - 1 units
E) ln( - 1) + 1 units

Question 38

A) - ln(2)
B) - ln(2)
C) + ln(2)
D) ln(2) -
E) - ln(32)

Question 39

Find the length of the arc y = ln(sec x) between x = 0 and x = .

A) ln(2 + ) units
B) ln( - 1) units
C) ln(1 + ) units
D) ln(2 - ) units
E) ln( ) units

Question 40

Find the arc length of the curve x = (y) from y = to y = .

A)
B)
C)
D)
E)

Question 41

Find the area of the surface obtained by rotating the curve y = , -1 $\le$ x $\le$ 1, about the y-axis.

A) (5 - 1) square units
B) (5 + 1) square units
C) (5 - 1) square units
D) (5 + 1) square units
E) (5 - 1) square units

Question 42

Find the area of the surface obtained by rotating the curve y = , -1 $\le$ x $\le$ 1, about the x-axis.

A) square units
B) square units
C) square units
D) square units
E) square units

Question 43

Find the area of the surface generated by rotating + = about y = a.

A) 6 square units
B) 8 square units
C) 2 square units
D) 4 square units
E) square units

Question 44

the x-axis.

Question 45

Find the area of the surface generated by rotating y = , - $\infty$ $\le$ x $\le$ 0 about y = 0.

A) $\pi$ square units
B) $\pi$ square units
C) $\pi$ square units
D) + ln(1 + ) square units
E) - ln(1 + ) square units

Question 46

Find the area of the oval surface obtained by rotating the ellipse + 4 = 1 about its major axis (i.e., about the x-axis).

A) - square units
B) - square units
C) + square units
D) + square units
E) - square units

Question 47

Find the length of the curve y = - from x = 0 to x = 3.

A) 2 units
B) units
C) 3 units
D) 4 units
E) 5 units

Question 48

Find the area of the surface generated by rotating y = - where x [0, 3] about y = 0.

A) 3 $\pi$ square units
B) 4 $\pi$ square units
C) 2 $\pi$ square units
D) 6 $\pi$ square units
E) $\pi$ square units

Question 49

Find, correct to 4 decimal places, the length of the curve y = from x = 1 to x = 8.

A) 19.1981 units
B) 14.6572 units
C) 3.4123 units
D) 7.6337 units
E) 22.8030 units

Question 50

Find the length of the closed loop part of the curve 3 = x .

A) units
B) units
C) units
D) 2 units
E) 1 unit

Question 51

Assuming the Earth is spherical with radius 6378 km, find the area of the surface of the Earth between the Tropic of Cancer (23.5° north latitude) and the Antarctic Circle (66.5° south latitude) as shown in the figure below.

Question 52

Find the centre of mass of the semicircular plate 0 $\le$ y $\le$ assuming it has constant density.

A)
B)
C)
D)
E)

Question 53

Find the centre of mass of a system of point masses m₁ = 6, m₂ = 3, m₃ = 2, and m₄ = 9 located at (3, -2), (0, 0), (-5, 3), and (4, 2), respectively.

A)
B)
C)
D)
E)

Question 54

A conical tank with vertex at the bottom and top radius 10 cm is 40 cm tall. It is filled with a substance whose density at depth y cm is (2 + y) g/ . Find the total mass of the substance filling the tank.

A) 15 $\pi$ kg
B) 16 $\pi$ kg
C) 17 $\pi$ kg
D) 18 $\pi$ kg
E) none of the above

Question 55

A triangular plate has vertices at (0, 0), (a, 0), and (0,b), where a > 0 and b > 0. The plate has variable thickness; at position (x, y) its thickness is . Assuming the plate is made of material of constant density, find the x-coordinate of its centre of mass.

Question 56

A triangular plate has vertices at (0, 0), (a, 0), and (0, b), where a > 0 and b > 0. The plate has variable thickness; at position (x, y) its thickness is . Assuming the plate is made of material of constant density, find the x-coordinate of its centre of mass.

A)
B)
C)
D)
E)

Question 57

Consider the finite plane region bounded by the coordinate axes and the line 4x + 3y = 12. Assuming the region has constant areal density 1, find the moments of this region about the coordinate axes.

A) = 8, = 6
B) = 6, = 8
C) = 6, = 6
D) = 8, = 8
E) = 6, = 0

Question 58

Find the mass of a thin plate that occupies the planar region described by 0 $\le$ y $\le$ sin(2x), 0 $\le$ x $\le$ if the areal density is given by (x) = 8x.

A) 32
B) 2
C) 4 $\pi$
D) $\pi$
E) 0

Question 59

Find the moment about the x-axis of a plate of constant areal density 1 occupying the finite plane region bounded by the x-axis and the curve y = -16 + 10x - .

A)
B)
C)
D)
E)

Question 60

Find the moment about the line x = 4 of a plate of constant density 1 occupying the finite plane region bounded by the x-axis and the curve y = -16 + 10x - x².

A) 32
B) 36
C) 34
D) 38
E) 30

Question 61

Find the moment about the x-axis of a thin plate that occupies the planar region described by 0 $\le$ y $\le$ , 0 $\le$ x $\le$ 1 if the areal density is given by (x) = e^x.

A)
B) 1
C) e
D) 2e - 1
E) e - 1

Question 62

A thin plate occupying the planar region 0 $\le$ x $\le$ g(y), c $\le$ y $\le$ d has mass equal to 2 units. If the areal density (y) = , = 6, and = 16, then the centre of mass of the plate is at the point:

A) ( , ) = (16, 12)
B) ( , ) = (12, 16)
C) ( , ) = (4, 3)
D) ( , ) = (0, 0)
E) ( , ) = (3, 4)

Question 63

Find the moment about the x-axis of the region in the first quadrant bounded by the lines y = 5x, y = 3x, x = 3. Assume the areal density is 1.

A) 16
B) 18
C) 17
D) 20
E) 14

Question 64

A hemispherical bowl of radius r cm is filled with water. How far below the surface is the centre of mass of this water?

A) cm
B) cm
C) cm
D) cm
E) cm

Question 65

The plane region defined by 0 ≤ y ≤ , 0 ≤ x ≤ a is revolved about the x-axis to generate a 3-dimensional region that is filled with material of constant density. Where is the centre of mass of this material?

Question 66

The plane region defined by 0 $\le$ y $\le$ , 0 $\le$ x $\le$ a is revolved about the x-axis to generate a 3-dimensional region that is filled with material of constant density. Where is the centre of mass of this material?

A) on the x-axis, units to the right of the origin
B) on the x-axis, units to the right of the origin
C) on the x-axis, units to the right of the origin
D) on the x-axis, units to the right of the origin
E) on the x-axis, units to the right of the origin

Question 67

Determine the centre of mass for the region bounded by y = 2 sin 2x, y = 0 on the interval [0, ].

A)
B)
C)
D)
E)

Question 68

Find the centroid of the region in the first quadrant bounded by the lines y = 5x, y = x, and x = 4.

A)
B)
C)
D)
E)

Question 69

Determine the centroid of the finite plane region bounded by y = x³ and y = .

A)
B)
C)
D)
E)

Question 70

Find the centroid of the region bounded by the x-axis and the curve y = -16 + 10 x - x².

A)
B)
C)
D)
E)

Question 71

Find the volume of the solid ring obtained by rotating the disc x² + = 9 about the x-axis.

A) 126 cubic units
B) 144 cubic units
C) 112 cubic units
D) 169 cubic units
E) 196 cubic units

Question 72

Find the centroid of the planar region bounded by y = , y = 0, x = 1, and x = 2.

A)
B)
C)
D)
E)

Question 73

Find the coordinates of the centroid of the region enclosed by y = sin(x) and the x-axis from x = 0 to .

A)
B)
C)
D)
E)

Question 74

Find the centroid of the finite plane region bounded by the curve y = 4 - x² and the liney = x + 2.

A)
B)
C)
D)
E)

Question 75

Find the centroid of the finite plane region bounded by y = x² and y = x.

A)
B)
C)
D)
E)

Question 76

A finite region R is contained in the first quadrant of the xy-plane. The centroid of R is the point (3, h). When R is revolved about the y-axis it generates a solid having volume 12 cubic units. When R is revolved about the x-axis it generates a solid having volume 32 cubic units. Find (a) the area of R and (b) the value of h.

A) (a) square units, (b) 8
B) (a) 2 square units, (b)
C) (a) square units, (b) 4
D) (a) 4 square units, (b)
E) (a) square units, (b) 4

Question 77

Find the centroid of the finite plane region bounded by y = , y = , and x = 1.

A)
B)
C)
D)
E)

Question 78

A triangle T has vertices ( , ), j = 1, 2, 3 (where each > 0). If the volume of the solid of revolution obtained by revolving T about the y-axis is 2 $\pi$ cubic units, what is the area of T?

A) square units
B) square units
C) square units
D) square units
E) square units

Question 79

Use Pappus's Theorem to find the volume of the solid of revolution obtained by rotating the triangular plane region specified by 0 $\le$ y $\le$ 1 - about (a) the line x = 2 and(b) the line y = 2.

A) (a) 4 $\pi$ , (b)
B) (a) 2 $\pi$ , (b)
C) (a) 4 $\pi$ , (b)
D) (a) 2 $\pi$ , (b)
E) (a) 2 $\pi$ , (b)

Question 80

Use Pappus's Theorem to find the volume of the solid generated by revolving the region R enclosed by y = , x = 0, and y =1 about the line y = -1 given that the centroid of the region R is at the point( , ) = ( , ).