Deck 15: Multiple Integration

Question 1

Use the Riemann sum corresponding to a subdivision of the rectangular region R defined by 0 $\le$ x $\le$ 6, 0 $\le$ y $\le$ 4, into six squares of edge length 2 and sample points at the upper-right corner of each square to estimate .

A) 72
B) 288
C) 144
D) 36
E) 168

Answer 1

288

Question 2

Use the Riemann sum corresponding to a subdivision of the rectangular region R defined by 0 $\le$ x $\le$ 6, 0 $\le$ y $\le$ 4, into six squares of edge length 2 and sample points at the centre of each square to estimate .

A) 72
B) 288
C) 144
D) 36
E) 120

Answer 2

144

Question 3

Evaluate dA, where R is the rectangle 6 $\le$ x $\le$ 9, -3 $\le$ y $\le$ 2, by interpreting it as a known volume.

A) 4
B) 15
C) 72
D) 60
E) 12

Answer 3

60

Question 4

Evaluate dA, where R is the square -2 $\le$ x $\le$ 2, -2 $\le$ y $\le$ 2.

A) 0
B) 2
C) $\pi$
D) 17.68
E) 2 $\pi$

Question 5

Evaluate dA, where R is the rectangle -2 $\le$ x $\le$ 2, -3 $\le$ y $\le$ 3, by interpreting it as a known volume.

A) 144
B) 48
C) 96
D) 24
E) 0

Question 6

Evaluate dA, where D is the disk + $\le$ 9, by interpreting it as a known volume.X

A) 12 $\pi$
B) 18 $\pi$
C) 24 $\pi$
D) 36 $\pi$
E) 9 $\pi$

Question 7

Evaluate , where T is the triangle in the xy-plane bounded by the coordinate axes and the line x + y = 1, by interpreting the double integral as a known volume.

A)
B)
C)
D)
E)

Question 8

Evaluate , where R is the rectangle 0 $\le$ x $\le$ 5, 2 $\le$ y $\le$ 10, by interpreting the double integral as a known volume.

A) 25
B) 50
C) 75
D) 100
E) 20

Question 9

Evaluate dA over the triangle T with vertices (0, 0), (0, 1), and (1,1).

A)
B)
C)
D)
E)

Question 10

Evaluate over the rectangle -1 $\le$ x $\le$ 2, 0 $\le$ y $\le$ 3.

A)
B)
C)
D)
E) 59

Question 11

Evaluate the double integral over the rectangle R bounded by the lines x = 0, x = 2, y = 1, and y = 2.

A)
B) ln 2
C) ln 2
D)
E) 2

Question 12

Evaluate , where R is the region bounded by y = 2x, y = 5x, and x = 2.

A) 144
B) 156
C) 160
D) 172
E) 184

Question 13

Evaluate , where R is the bounded region bounded by y = x and y = x².

A)
B)
C)
D)
E)

Question 14

Evaluate where R is the planar region described by 0 ≤ x ≤ 0 ≤ y ≤ cos(x).

A) 4
B) 12
C) 0
D) - 3
E) 8

Question 15

Evaluate the double integral where T is the quadrilateral bounded by the lines x = 1, x = 2, y = x, and y =

A) ln 2
B) ln 2
C) ln 2
D) ln 2
E) ln 2

Question 16

Evaluate the iterated integral dx dy by first reiterating it in the opposite direction.

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

Question 17

Evaluate dA, where T is the triangle with vertices (0, 0), (0, 2), and (2, 2).

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

Question 18

Evaluate dA, where R is the region defined by the inequalities x² $\le$ y $\le$ x.

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

Question 19

Find the volume of the solid bounded above by the paraboloid z = 9x² + y², below by the plane , and laterally by the planes x = 0, y = 0, x = 3, and y = 2.

A) 170 cubic units
B) 180 cubic units
C) 190 cubic units
D) 200 cubic units
E) 90 cubic units

Question 20

Evaluate dA, where D is the rectangular region described by the inequalities 0 $\le$ x $\le$ 10 ln(13), 25 $\le$ y $\le$ 50.

A) 144
B) 1
C)
D) 143
E)

Question 21

Find the volume of the solid in the first octant inside the cylinder x² + y² = 2y and under the plane z = 2 - x.

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

Question 22

Evaluate the iterated integral dy by first reversing the order of the integration.

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

Question 23

Find the volume of the solid lying inside the cylinder x² + y² = 4, above the plane z = x - y - 8, and below the surface z = 8 - x³.

A) 72 $\pi$ cubic units
B) 64 $\pi$ cubic units
C) 48 $\pi$ cubic units
D) 32 $\pi$ cubic units
E) 56 $\pi$ cubic units

Question 24

Evaluate where D is the region in the xy-plane enclosed by the parallelogram with vertices at the points (2, 4), (4, 9), (10, 9), and (8, 4).

A) 12
B) 15
C) 0
D) 6
E) 30

Question 25

Find the volume of the solid below the surface z = 3y² and above the triangular region in the xy-plane bounded by the straight lines x = 0, y = 0, and x + 2y = 2.

A) 6
B) 2
C)
D) 12
E)

Question 26

The iterated integral is the double integral of g(x,y) over a planar region R.
(i) Sketch the planar region R.
(ii) Express the double integral J as a sum of two iterated integrals with the order of the integrals reversed.

Question 27

Find the volume of the solid bounded by the paraboloid z = 16 - x² - 4y² and the planez = 0.

A) 48 $\pi$ cubic units
B) 64 $\pi$ cubic units
C) 96 $\pi$ cubic units
D) 128 $\pi$ cubic units
E) 81 $\pi$ cubic units

Question 28

The planar region enclosed by the straight lines y = x, y = 1 + x, y = -x, and y = 1 -x is both x-simple and y-simple.

Question 29

Evaluate the double integral dx dy, where Q is the first quadrant of the xy-plane.

A)
B) 15
C) 1
D) $\infty$ (integral diverges)
E) -

Question 30

Evaluate the double integral dx dy, where R is the region under the curve xy = 1, above the x-axis, and to the right of the line x = 1.

A)
B)
C)
D) $\infty$ (integral diverges)
E)

Question 31

Evaluate the double integral dx dy, where S is the part of the first quadrant of the xy-plane lying above the line x = 2y.

A)
B)
C)
D) $\infty$ (integral diverges)
E) 1

Question 32

Evaluate dx dy, where S is the semi-infinite strip 0 $\le$ x $\le$ 1, 0 $\le$ y < $\infty$ .

A) 1
B)
C)
D) $\infty$ (integral diverges)
E) 2

Question 33

Evaluate where T is the triangle with vertices (0, 0), (1, 1), and (1, 2).

A) ln(2)
B) 2 ln(2)
C) 2
D) $\infty$ (integral diverges)
E)

Question 34

Evaluate where T is the triangle with vertices (0, 0), (1, 1), and (1, 2).

A) ln(2)
B) 2 ln(2)
C) 2
D) $\infty$ (integral diverges)
E)

Question 35

dA, where T is the triangular region enclosed by the straight lines y = -x, y = 7x, and y = x + 6 is an improper integral.

Question 36

The improper double integral dA, where D is the planar region described by1 $\le$ x < $\infty$ , 0 $\le$ y $\le$ x⁸, converges for all real numbers k such that

A) k $\ge$ - 7
B) k > 5
C) k < 9
D) k <- 7
E) k $\le$ 5

Question 37

Find if R is the unit circular disk x² + y² $\le$ 9.

A) 9
B) 18
C)
D) 18 $\pi$
E) 9 $\pi$

Question 38

Find the average value of x² + y² over the disk x² + y² $\le$ 4.

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

Question 39

Find if R is the unit circular disk.

A)
B)
C)
D)
E)

Question 40

Find where R is the region in the first quadrant lying between the circles x² + y² = 1 and x² + y² = 4 and between the lines y = 0 and y = x.

A) ln 2
B) ln 2
C) ln 2
D) ln 2
E) ln 2

Question 41

Find the volume of the given solid by transforming to a double integral in polar coordinates.The solid is bounded by x² + y² = 36, z = 0, and z = x² + y².

A) 618 $\pi$ cubic units
B) 628 $\pi$ cubic units
C) 638 $\pi$ cubic units
D) 648 $\pi$ cubic units
E) 658 $\pi$ cubic units

Question 42

Use polar coordinates to find the volume of the solid lying under the cone z = , above the plane z = 0, and inside the cylinder x² + y² = 2x.

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

Question 43

Find the volume remaining after a cylindrical hole of radius b is drilled through the centre of a ball of radius a > b.

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

Question 44

Use polar coordinates to evaluate .

A)
B)
C)
D)
E)

Question 45

Expressed in polar coordinates, the area enclosed by a planar region D is equal to dr d $\theta$ .

Question 46

Find dA, where C is the cardioid disk 0 $\le$ r $\le$ 1 + cos $\theta$ .

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

Question 47

Find the volume of the solid of revolution obtained by rotating the plane region bounded by the cardioid r = 1 + sin $\theta$ about the y-axis.

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

Question 48

Evaluate the iterated integral by first transforming it to an iterated integral in polar coordinates.

A)
B)
C)
D)
E) 243 $\pi$

Question 49

Evaluate the integral using polar coordinates.

A)
B)
C)
D)
E)

Question 50

Find dA, where R is the region outside the circle r = 1 and inside the cardioid r = 1 + cos $\theta$ .

A)
B)
C)
D)
E)

Question 51

If I = , then I = , and so = , where R² is the entire xy-plane. Evaluate this double integral by iterating it in polar coordinates and hence find the value of I.

A) I =
B) I = $\pi$
C) I =
D) I =
E) I = 2 $\pi$

Question 52

Use polar coordinates to find the volume of the solid enclosed by the surfaces z = 3 and .

A) 16 $\pi$
B) $\pi$
C) 4 $\pi$
D) 8 $\pi$
E) 7 $\pi$

Question 53

Let R be the region in the first quadrant of the xy-plane bounded by the curves xy = 1 and xy = 4 and the lines y = x and y = 2x. Use a suitable coordinate transformation to evaluate and hence find the area of R.

A) 8 square units
B) 4 ln(3) square units
C) 5 ln(3) square units
D) 3 ln(2) square units
E) 16 square units

Question 54

Use a suitable change of variables to evaluate

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

Question 55

Use the transformation u = x + y, v = to evaluate the double integral of f(x, y) = over the triangular region bounded by the lines x + y = 1, x = 0, and y = 0.

A)
B)
C)
D)
E)

Question 56

Evaluate dV, where R is the rectangular box 0 $\le$ x $\le$ 1, 1 $\le$ y $\le$ 2, 1 $\le$ z $\le$ 2.

A)
B)
C)
D)
E)

Question 57

Compute the integral where R is the tetrahedral region bounded by the planes x = 0, y = 0, z = 0, and x + y + z = 4.

A) 16
B) 8
C) 32
D) 24
E)

Question 58

Evaluate the integral dV, where R is the region defined by the inequalities0 $\le$ y $\le$ 1, 0 $\le$ z $\le$ y, 0 $\le$ x $\le$ z.

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

Question 59

Evaluate the integral dV, where R is the region in the first octant bounded by the four planes x = 1, y = x, z = 0, and z = y.

A) 3
B) 2
C) 1
D) 4
E) 5

Question 60

Evaluate the triple integral over the region where R is the finite region bounded by z = 0, y + z = 4, y = x².

A)
B)
C)
D)
E)

Question 61

Express the iterated integral as an equivalent integral in which the outermost integral is with respect to x and the innermost is with respect to y.

A)
B)
C)
D)
E)

Question 62

Evaluate where E is the region in 3-space described by the inequalities0 ≤ x ≤ 2 - y - z, 0 ≤ z ≤ 2 - y, and 0 ≤ y ≤ 2.

Question 63

Evaluate by completely reversing the order in which the integrals are performed.

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

Question 64

Find the volume of the solid that lies below the surface z = 1 + and above the region in the xy-plane bounded by the straight lines y = -x, y = x, and y = 1.

A) V =
B) V =
C) V =
D) V =
E) V =

Question 65

Use a triple integral to find the volume V of the solid inside the cylinder x² + y² = 16 and between the planes z = x - y - 2 and z = 6 + x - y.

A) 64 $\pi$ cubic units
B) 32 $\pi$ cubic units
C) 128 $\pi$ cubic units
D) 256 $\pi$ cubic units
E) 144 $\pi$ cubic units

Question 66

Evaluate the iterated integral by transforming it to cylindrical coordinates.

A) 5 $\pi$
B) 4 $\pi$
C) 3 $\pi$
D) 2 $\pi$
E) $\pi$

Question 67

Evaluate where E is the region in space enclosed by the sphere(x - 1)² + y² + z² = 3.

A) 9 $\pi$
B) 12 $\pi$
C) 27
D) 36 $\pi$
E) 4 $\pi$

Question 68

Evaluate the triple integral where R is the first octant region abovez = x² + y² and below

A)
B)
C)
D)
E)

Question 69

Use a triple integral iterated in spherical coordinates to find the volume of the region lying above the cone and inside the sphere x² + y² + z² = a².

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

Question 70

Evaluate dV, where C is the right circular cylinder consisting of all points (x, y, z) satisfying x² + y² $\le$ 3 and -1 $\le$ z $\le$ 2.

A) 3^3/2
B) 3^-(3/2) $\pi$
C) 3^5/2
D) 3^3/2 $\pi$
E) 27 $\pi$

Question 71

The plane x + y + z = 1 slices the ball x² + y² + z² $\le$ 1 into two pieces. Find the volume of the smaller piece. (Hint: Replace the plane by a horizontal plane at the same distance from the origin.)

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

Question 72

Use the transformation x = au, y = bv, z = cw (where a, b, c are positive constants) to evaluate the triple integral of z over the first-octant region bounded by the coordinate planes and the ellipsoid

A)
B)
C) ac
D) bc
E) $\pi$ ab

Question 73

Evaluate the iterated integral by transforming it to cylindrical coordinates.

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

Question 74

Evaluate dV, a > 0, where R is the right circular cylinder consisting of all points (x, y, z) satisfying x² + y² $\le$ b, b > 0 and -3 $\le$ z $\le$ 5 by using cylindrical coordinates.

A) 4 ln
B) 8 ln
C) 4 $\pi$ ln
D) 8 $\pi$ ln
E) 2 $\pi$ ln

Question 75

A solid S has the shape of the region E enclosed by the sphere R = 3cos( ). Evaluate dV .
Note: R ,11ee7b54_ddf4_a0dd_ae82_79cd253742f9_TB9661_11 , $\theta$ are the spherical coordinates.

A) 12
B) 3 $\pi$
C) 6
D) 9
E)

Question 76

Use cylindrical coordinates to compute dV where E is the region bounded by the paraboloid z = 6x² + 4y² and the cylinder z = 6 - 2y².

Question 77

Evaluate where E is the region enclosed by the ellipsoid + + = 1, a, b, c > 0. Use the transformation x = au, y = bv, z = cw.

A) ab
B) ab
C) ab
D) ab
E) ab

Question 78

Let J = where E is the region enclosed by the paraboloids z = 2( + ) and . Express J in cylindrical coordinates [r, $\theta$ , z]. Do not evaluate.

A)
B)
C)
D)
E)

Question 79

Evaluate , where R is the region x² + y² + z² $\le$ 4, $\le$ z² $\le$ 3(x² + y²), z $\le$ 0.(Hint: Use spherical coordinates.)

A) 8 $\pi$
B) 2 $\pi$
C) 16 $\pi$
D) 4 $\pi$
E) 32 $\pi$

Question 80

Evaluate dV, where B is the ball x² + y² + z² $\le$ a², a > 0.

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