Deck 16: Integrating Functions of Several Variables

Question 1

Set up but do not evaluate a (multiple)integral that gives the volume of the solid bounded above by the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 2$ and below by the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ .

A) $\int _ { - 1 } ^ { 1 } \int _ { - \sqrt { 1 - x ^ { 2 } } } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { x ^ { 2 } + y ^ { 2 } } ^ { \sqrt { 2 - x ^ { 2 } - y ^ { 2 } } } d x d y d z$
B) $\int _ { - 1 } ^ { 1 } \int _ { - \sqrt { 1 - x ^ { 2 } } } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { - x ^ { 2 } - y ^ { 2 } } ^ { \sqrt { 2 - x ^ { 2 } - y ^ { 2 } } } d z d y d x$
C) $$\int_{-1}^{1} \int_{0}^{\sqrt{1-x^{2}}} \int_{x^{2}+y^{2}}^{\sqrt{2-x^{2}-y^{2}}}$$$d z d y d x$
D) $\int _ { - 1 } ^ { 1 } \int _ { - \sqrt { 1 - x ^ { 2 } } } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { x ^ { 2 } + y ^ { 2 } } ^ { \sqrt { 2 - x ^ { 2 } - y ^ { 2 } } } d z d y d x$
E) $\int _ { 0 } ^ { 1 } \int _ { - \sqrt { 1 - x ^ { 2 } } } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { x ^ { 2 } + y ^ { 2 } } ^ { \sqrt { 2 - x ^ { 2 } - y ^ { 2 } } } d z d y d x$

Answer 1

$\int _ { - 1 } ^ { 1 } \int _ { - \sqrt { 1 - x ^ { 2 } } } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { x ^ { 2 } + y ^ { 2 } } ^ { \sqrt { 2 - x ^ { 2 } - y ^ { 2 } } } d z d y d x$

Question 2

Estimate $\int$ _R f(x, y)dA using the table of values below, where R is the rectangle 0 $\le$ x $\le$ 4, 0 $\le$ y $\le$ 6
$\begin{array}{c}y\\x\begin{array}{cccc}&0 & 3 & 6\\0 & 2 & 3 & 4 \\2 & 6 & 4 & 3 \\4 & 18 & 14 & 12\end{array}\end{array}$

Answer 2

165

Question 3

Upper and lower sums for a function f on a rectangle R, using n subdivisions on each side, are and respectively.Evaluate

Answer 3

3

Question 4

Consider the integral $\int _ { 1 } ^ { 5 } \int _ { 9 x } ^ { 45 } f ( x , y ) d y d x$ Rewrite the integral with the integration performed in the opposite order.

A) $\int _ { 9 x } ^ { A 5 } \int _ { 1 } ^ { y / 9 } f ( x , y ) d x d y$
B) $\int _ { 9 } ^ { 4 5 } \int _ { 1 } ^ { y / 9 } f ( x , y ) d y d x$
C) $\int _ { 9 } ^ { 4 5 } \int _ { 1 } ^ { y / 9 } f ( x , y ) d x d y$
D) $\int _ { 9 } ^ { 45 } \int _ { 1 } ^ { 0 y } f ( x , y ) d x d y$

Question 5

Evaluate the iterated integral.

Question 6

Calculate the following integral exactly.(Your answer should not be a decimal approximating the true answer, but should be exactly equal to the true answer.Your answer may contain e, $\pi$ , $\sqrt { 2 }$ , and so on.) $\int _ { 0 } ^ { 6 } \int _ { 0 } ^ { 3 } \cos 4 y \sin ( 4 x + 5 ) d x d y$

Question 7

Let R be the region in the first quadrant bounded by the x- and y-axes and the line x + y = 7.Evaluate exactly and then give an answer rounded to 4 decimal places.

Question 8

Evaluate the iterated integral.

Question 9

Reverse the order of integration for the following integral. $\int _ { 1 } ^ { 3 } \int _ { x ^ { 2 } - 9 } ^ { 4 x - 12 } f ( x , y ) d y d x$

A) $\int _ { 0 } ^ { 8 } \int _ { ( y + 12 ) / 4 } ^ { \sqrt { y + 9 } } f ( x , y ) d x d y$
B) $\int _ { - 8 } ^ { 0 } \int _ { ( y + 12 ) / 4 } ^ { \sqrt { y + 9 } } f ( x , y ) d y d x$
C) $\int _ { - 8 } ^ { 0 } \int _ { ( y + 12 ) / 4 } ^ { \sqrt { y + 9 } } f ( x , y ) d x d y$
D) $\int _ { - 8 } ^ { 0 } \int _ { \sqrt { y + 9 } } ^ { ( y + 12 ) / 4 } f ( x , y ) d x d y$
E) $\int _ { - 8 } ^ { 0 } \int _ { ( y - 12 ) / 4 } ^ { \sqrt { y + 9 } } f ( x , y ) d x d y$

Question 10

Evaluate by first reversing the order of integration.

Question 11

Let f(x, y)be a positive function of x and y which is independent of x, that is, f(x, y)= g(y)for some one-variable function g.Suppose that $\int _ { 0 } ^ { 3 } g ( x ) d x = 10$ and $\int _ { 0 } ^ { 10 } g ( x ) d x = 1$ .
Find $\int _ { R } f d A$ , where R is the rectangle 0 $\le$ x $\le$ 3, 0 $\le$ y $\le$ 10.

Question 12

Let R₁ be the region 0 $\le$ x $\le$ 3, -2 $\le$ y $\le$ 4, and let R₂ be the region 3 $\le$ x $\le$ 5, -2 $\le$ y $\le$ 4.Suppose that the average value of f over R₁ is 6 and the average value over R₂ is 7.
Find the average value of f over R, $0 \leq x \leq 5 , - 2 \leq y \leq 2$ .

Question 13

True or false?
If f is any two-variable function, then $\int _ { R } f d A = 2 \int _ { S } f d A$ , where R is the rectangle 0 $\le$ x $\le$ 2, 0 $\le$ y $\le$ 1 and S is the square 0 $\le$ x, y $\le$ 1.

Question 14

Find the volume of the region under the graph of and above the region

Question 15

Find a region R such that double integral $\int _ { R } \left( 2 - x ^ { 2 } - y ^ { 2 } \right) d A$ has the largest value.

A) $x ^ { 2 } - y ^ { 2 } \leq 2$
B) $x ^ { 2 } - y ^ { 2 } \geq 2$
C) $x ^ { 2 } + y ^ { 2 } \leq 2$
D) $x ^ { 2 } + y ^ { 2 } \geq 2$

Question 16

Find the volume under the graph of lying over the triangle with vertices (0, 0), (2, 2)and (4, 0).

Question 17

Reverse the order of integration for the following integral. $\int _ { 0 } ^ { 2 } \int _ { y ^ { 2 } } ^ { \sqrt { 8 y } } f ( x , y ) c ^ { 2 } x d y$

A) $\int _ { 0 } ^ { 2 } \int _ { x ^ { 2 } / 8 } ^ { \sqrt { x } } f ( x , y ) d y d x$
B) $\int _ { 0 } ^ { 4 } \int _ { x ^ { 2 } / 8 } ^ { \sqrt { x } } f ( x , y ) d x d y$
C) $\int _ { 0 } ^ { 4 } \int _ { x ^ { 2 } / 8 } ^ { x } f ( x , y ) d y d x$
D) $\int _ { 0 } ^ { 4 } \int _ { 8 x ^ { 2 } } ^ { \sqrt { x } } f ( x , y ) d y d x$
E) $\int _ { 0 } ^ { 4 } \int _ { x ^ { 2 } / 8 } ^ { \sqrt { x } } f ( x , y ) d y d x$

Question 18

Calculate the following integral exactly.(Your answer should not be a decimal approximating the true answer, but should be exactly equal to the true answer.Your answer may contain e, $\pi$ , $\sqrt { 2 }$ , and so on.) $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { z } \int _ { 0 } ^ { y } x ^ { 2 } y ^ { 5 } z ^ { 5 } d x d y d z$

Question 19

The function has an average value of 4 on the triangle with vertices at (0, 0), (0, 1)and (1, 0).Find the constant a.

Question 20

Calculate the following integral exactly.(Your answer should not be a decimal approximating the true answer, but should be exactly equal to the true answer.Your answer may contain e, $\pi$ , $\sqrt { 2 }$ , and so on.) $\int _ { 3 } ^ { 4 } \int _ { 0 } ^ { y } y ^ { 2 } e ^ { x y } d x d y$

A) $\frac { 1 } { 2 } \left( - 7 - e ^ { 9 } + e ^ { 16 } \right)$
B) $\frac { 1 } { 2 } \left( 25 - e ^ { 9 } + e ^ { 16 } \right)$
C) $\frac { 1 } { 2 } \left( - 7 + e ^ { 9 } - e ^ { 16 } \right)$
D) $\frac { 1 } { 3 } \left( - 7 - e ^ { 9 } + e ^ { 16 } \right)$

Question 21

Set up an iterated integral for $\int _ { W } f ( x , y , z ) d V$ , where W is the solid region bounded below by the rectangle 0 $\le$ x $\le$ 3, 0 $\le$ y $\le$ 1 and above by the surface $z ^ { 2 } + y ^ { 2 } = 1$

A) $\int _ { W } f ( x , y , z ) d V = \int _ { 0 } ^ { 3 } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - y ^ { 2 } } } f ( x , y , z ) d z c x \alpha d y$
B) $\int _ { W } f ( x , y , z ) d V = \int _ { 0 } ^ { 3 } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - y ^ { 2 } } } f ( x , y , z ) d y d z d x$
C) $\int _ { W } f ( x , y , z ) d V = \int _ { 0 } ^ { 3 } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - y ^ { 2 } } } f ( x , y , z ) d z d y d x$
D) $\int _ { W } f ( x , y , z ) d V = \int _ { 0 } ^ { 3 } \int _ { 0 } ^ { 1 } \int _ { - \sqrt { 1 - y ^ { 2 } } } ^ { \sqrt { 1 - y ^ { 2 } } } f ( x , y , z ) d z d y d x$
E) $\int _ { W } f ( x , y , z ) d V = \int _ { 0 } ^ { 3 } \int _ { 0 } ^ { 1 } \int _ { - \sqrt { 1 - y ^ { 2 } } } ^ { \sqrt { 1 - y ^ { 2 } } } f ( x , y , z ) d y d x d z$

Question 22

A cylindrical tube of radius 2cm and length 3cm contains a gas.As the tube spins around its axis, the density of the gas increases as you get farther from the axis.The density, D, at a distance of r cm from the axis is D(r)= 1 +9 r gm/cc.
Write a triple integral representing the total mass of the gas in the tube and evaluate the integral.

Question 23

Set up (but do not evaluate)an iterated integral to compute the mass of the solid paraboloid bounded by $z = x ^ { 2 } + y ^ { 2 }$ and z = 1, if the density is given by $\delta$ (x, y, z)= z².

A) $\text { Mass } = \int _ { - 1 } ^ { 1 } \int _ { - \sqrt { 1 - x ^ { 2 } } } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { x ^ { 2 } + y ^ { 2 } } ^ { 1 } z ^ { 2 } d z d x d y$
B) $\text { Mass } = \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { x ^ { 2 } + y ^ { 2 } } ^ { 1 } z ^ { 2 } d z d y d x$
C) $\operatorname { Mass } = \int _ { - 1 } ^ { 1 } \int _ { - \sqrt { 1 - x ^ { 2 } } } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { 0 } ^ { 1 } z ^ { 2 } d z d y d x$
D) $\text { Mass } = \int _ { - 1 } ^ { 1 } \int _ { - \sqrt { 1 - x ^ { 2 } } } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { x ^ { 2 } + y ^ { 2 } } ^ { 1 } z ^ { 2 } d z d y d x$
E) $\text { Mass } = \int _ { 0 } ^ { 1 } \int _ { - \sqrt { 1 - x ^ { 2 } } } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { x ^ { 2 } + y ^ { 2 } } ^ { 1 } z ^ { 2 } d z d y d x$

Question 24

Compute the area of the flower-like region bounded by r = 6 + 3 cos (8 $\theta$ ).

Question 25

Find the triple integral of the function f(x, y, z)= xy sin (18yz)over the rectangular box 0 $\le$ x $\le$ $\pi$ , 0 $\le$ y $\le$ 1, 0 $\le$ z $\le$ $\pi$ /6.

Question 26

Set up the three-dimensional integral $\int _ { R } y d V$ where R is the "ice-cream cone" enclosed by a sphere of radius 2 centered at the origin and the cone $z = \sqrt { 3 x ^ { 2 } + 3 y ^ { 2 } }$ .Use rectangular coordinates.

A) $\int_{R} y d \mathrm{~V}=\int_{0}^{2 \pi} \int_{0}^{1} \int_{\sqrt{3 r}}^{\sqrt{4-r^{2}}} r^{2} \sin \theta d z d r d \theta$
B) $\int _ { R } y d V = \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { \pi / 6 } \int _ { 0 } ^ { 2 } \rho ^ { 3 } \sin \theta \sin ^ { 2 } \phi d \rho d \phi d \theta$
C) $\int _ { R } y d V = \int _ { - 1 } ^ { 1 } \int _ { - \sqrt { 1 - x ^ { 2 } } } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { \sqrt { 3 x ^ { 2 } + 3 y ^ { 2 } } } ^ { \sqrt { 4 - x ^ { 2 } - y ^ { 2 } } } y d z d y d x$

Question 27

Consider the volume between a cone centered along the positive z-axis, with vertex at the origin and containing the point (0, 1, 1), and a sphere of radius 3 centered at the origin.
Write a triple integral which represents this volume and evaluate it.Use spherical coordinates.

Question 28

Calculate the following integral:

Question 29

Calculate the following integral: where R is the shaded region shown below.
·

Question 30

Suppose a solid is the region in three-space in the first octant bounded by the plane x + y = 1 and the cylinder .If the density of this solid at a point (x, y, z)is given by , find its mass.

Question 31

A solid is bounded below by the triangle z = 0, x $\ge$ 0, y $\ge$ 0, x + y $\le$ 1 and above by the plane z = x + 6y + 2.If the density of the solid is given by $\delta$ (x, y, z)= z, find its mass.

Question 32

Evaluate Provide an exact answer.

Question 33

For the following region, decide whether to integrate using polar or Cartesian coordinates.Write an iterated integral of an arbitrary function f(x, y)over the region.

Question 34

Evaluate the integral , where R is the region in the first quadrant bounded by the y-axis, the line y = x and the circles

Question 35

Let R be the ice-cream cone lying inside the sphere and inside the cone .Find the center of mass of R.

Question 36

Evaluate exactly the integral , where R is the region shown below.

Question 37

Sketch the region of integration of the following integral and then convert the expression to polar co-ordinates (you do not have to evaluate it).

Question 38

Evaluate the iterated integral

Question 39

Find the volume of the solid bounded by the paraboloid and the plane z = 1.

Question 40

Convert the integral to polar coordinates and hence evaluate it exactly.

Question 41

Let x and y have joint density function Find the probability that x > y +0.4.

Question 42

Choose the most appropriate coordinate system and set up a triple integral, including limits of integration, for a density function f(x, y, z)over the given region.

A)spherical coordinates
B)cylindrical coordinates
C)rectangular coordinates
D)None of the above.

Question 43

Evaluate the integral in spherical coordinates.

Question 44

Evaluate where W is the first octant portion of the ball of radius 3 centered at the origin.

Question 45

Let W be the region between the cylinders and in the first octant and under the plane z = 1.Evaluate

Question 46

Rewrite the integral in spherical coordinates.You do not have to evaluate the integral.

Question 47

Evaluate the integral by interchanging the order of integration. .

Question 48

Let x and y have joint density function $p ( x , y ) = \left\{ \begin{array} { l l } x + y & \text { if } 0 \leq x \leq 1,0 \leq y \leq 1 \\0 & \text { otherwise }\end{array} \right.$ Find the probability that 0.5 $\le$ x $\le$ 0.6.

Question 49

Consider the change of variables x = s + 3t, y = s - 2t.
Let R be the region bounded by the lines 2x + 3y = 1, 2x + 3y = 4, x - y = -3, and x - y = 2.Find the region T in the st-plane that corresponds to region R.
Use the change of variables to evaluate $\int _ { R } 2 x + 3 y d A$

Question 50

The joint density function for x, y is given by Find the probability that (x, y)satisfies

Question 51

An arrow strikes a circular target at random at a point (x, y), using a coordinate system with origin at the center of the target.The probability density function for the point where the arrow strikes is given by What is the probability that the arrow strikes within 0.45 feet of the center of the target.

Question 52

Find the mass of the solid cylinder with density function

Question 53

Evaluate the integral .Give your answer to two decimal places.

Question 54

Find the condition on the non-negative constants a and b for p(x, y)to be a joint density function, where $p ( x , y ) = \left\{ \begin{array} { c l } a x + b y , & 0 \leq x \leq 8,0 \leq y \leq 8 \\0 & \text { otherwise }\end{array} \right.$

A) $a + b = \frac { 1 } { 256 }$
B) $a + b = \frac { 1 } { 32 }$
C) $a + b < \frac { 1 } { 256 }$
D)a=b
E) $a + b = \frac { 1 } { 512 }$

Question 55

Consider the change of variables x = s + 5t, y = s - 3t.
Find the absolute value of the Jacobian .

Question 56

Let R be the region bounded between the two ellipses $\frac { x ^ { 2 } } { 3 ^ { 2 } } + \frac { y ^ { 2 } } { 2 ^ { 2 } } = 1$ and $\frac { x ^ { 2 } } { 3 ^ { 2 } } + \frac { y ^ { 2 } } { 2 ^ { 2 } } = 4$ Use this change of coordinates $x=3 r \cos t, y=2 r \sin t$ for $r \geq 0,0 \leq t \leq 2 \pi$ to evaluate the integral $\int _ { R } \left( 4 x ^ { 2 } + 9 y ^ { 2 } \right) d A$

A)240 $\pi$
B)3240 $\pi$
C)162 $\pi$
D)1620 $\pi$
E)1620

Question 57

Jane and Mary will meet outside the library at noon.Jane's arrival time is x and Mary's arrival time is y, where x and y are measured in minutes after noon.The probability density function for the variation in x and y is After Jane arrives, she will wait up to 15 minutes for Mary, but Mary won't wait for Jane.Find the probability that they meet.

Question 58

Evaluate the integral in cylindrical coordinates.

Question 59

The region W is shown below.Write the limits of integration for $\int _ { w } f ( x , y , z ) d V$ in spherical coordinates.

A) $\int _ { W } f ( x , y , z ) d V = \int _ { 0 } ^ { \pi } \int _ { \pi / 2 } ^ { \pi } \int _ { 0 } ^ { 1 } f ( \rho \sin \phi \cos \theta , \rho \sin \phi \sin \theta , \rho \cos \phi ) \rho ^ { 2 } \sin \phi d \rho d \phi d \theta$
B) $\int _ { W } f ( x , y , z ) d V = \int _ { - 1 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { - \sqrt { 1 - x ^ { 2 } - y ^ { 2 } } } ^ { 0 } f ( x , y , z ) d z d y d x$
C) $\int _ { W } f ( x , y , z ) d V = \int _ { 0 } ^ { \pi } \int _ { 0 } ^ { 1 } \int _ { - \sqrt { 1 - r ^ { 2 } } } ^ { 0 } f ( r \cos \theta , r \sin \theta , z ) r d z d r d \theta$

Question 60

Let R be the region in the first quadrant bounded between the circle and the two axes.Then Let be the region in the first quadrant bounded between the ellipse and the two axes.
Use the change of variable x = s/5, y = t/3 to evaluate the integral

Question 61

The joint density function for random variables x and y is Find the probability .Give your answer to 3 decimal places.

Question 62

If f and g are two continuous functions on a region R, then $\int _ { R } f \cdot g d A = \int _ { R } f d A \cdot \int _ { R } g d A$ .

Question 63

Consider the integral .
(a)Sketch the region of integration and rewrite the integral with order of integration reversed.
(b)Rewrite the integral in polar coordinates.

Question 64

The joint density function for x, y is given by $p ( x , y ) = \left\{ \begin{array} { c c } \frac { 1 } { 150 } e ^ { - x / 10 } e ^ { - y / 15 } & x \geq 0 , y \geq 0 \\0 & \text { otherwise }\end{array} \right.$ Write down an iterated integral to compute the probability that x + y $\le$ 10.You do not need to do the integral.

A) $\int _ { 0 } ^ { 10 } \int _ { 0 } ^ { 10 } \frac { 1 } { 150 } e ^ { - x / 10 } e ^ { - y / 15 } d ^ { } y d x$
B) $\int _ { 0 } ^ { 10 } \int _ { 0 } ^ { 10 - x } \frac { 1 } { 150 } e ^ { - x / 10 } e ^ { - y / 15 } d x d y$
C) $\int _ { 0 } ^ { 10 } \int _ { 0 } ^ { 10 - x } \frac { 1 } { 150 } e ^ { - x / 10 } e ^ { - y / 15 } d y d x$
D) $\int _ { 0 } ^ { 10 } \int _ { 0 } ^ { 10 - x } \frac { 1 } { 10 } e ^ { - x / 10 } e ^ { - y / 15 } d y d ^ { 2 } x$
E) $\int _ { 0 } ^ { 10 - x } \int _ { 0 } ^ { 10 } \frac { 1 } { 150 } e ^ { - x / 10 } e ^ { - y / 15 } d x d y$

Question 65

Let W be the part of the solid sphere of radius 4, centered at the origin, that lies above the plane z = 2.Express $\int _ { W } z d V$ in
(a)Cartesian
(b)Cylindrical
(c)Spherical coordinates.

Question 66

Consider the integral .Convert the integral to polar coordinates.

Question 67

Evaluate the integral , where R is the region shown below.

Question 68

Consider the region in 3-space bounded by the surface and the plane where .Find the value of k such that the volume of this region below the xy-plane equals the volume of this region above the xy-plane.

Question 69

Convert the integral to polar coordinates. $\int _ { - 1 } ^ { 1 } \int _ { 1 } ^ { \sqrt { 2 - x ^ { 2 } } } f ( x , y ) d y d x$

A) $\int _ { \pi / 4 } ^ { 3 \pi / 4 } \int _ { 1 / \sin \theta } ^ { \sqrt { 2 } } f ( r \cos \theta , r \sin \theta ) d r d \theta$
B) $\int _ { 0 } ^ { 3 \pi / 4 } \int _ { 1 / \sin \theta } ^ { \sqrt { 2 } } f ( r \cos \theta , r \sin \theta ) r d r d \theta$
C) $\int _ { \pi / 4 } ^ { 3 \pi / 4 } \int _ { 1 } ^ { \sqrt { 2 } } f ( r \cos \theta , r \sin \theta ) r d r d \theta$
D) $\int _ { \pi / 4 } ^ { 3 \pi / 4 } \int _ { 1 / \sin \theta } ^ { \sqrt { 2 } } f ( r \cos \theta , r \sin \theta ) r d r d \theta$
E) $\int _ { \pi / 4 } ^ { 3 \pi / 4 } \int _ { 1 / \sin \theta } ^ { \sqrt { 2 } } f ( r \cos \theta , r \sin \theta ) r d \theta d r$

Question 70

Evaluate the integral in spherical coordinates.

Question 71

Find the mass of the solid cylinder , with density function

Question 72

Find the area of the part of the hyperbolic paraboloid $z = y ^ { 2 } - x ^ { 2 }$ that lies between the cylinders $x ^ { 2 } + y ^ { 2 } = 1$ and $x ^ { 2 } + y ^ { 2 } = 16$ .

A) $\frac { 65 \sqrt { 65 } - 5 \sqrt { 5 } } { 3 } \pi$
B) $\frac { \sqrt { 65 } - \sqrt { 5 } } { 6 } \pi$
C) $\frac { 65 \sqrt { 65 } - 5 \sqrt { 5 } } { 6 } \pi$
D) $\frac { \sqrt { 65 } - \sqrt { 5 } } { 3 } \pi$
E) $\frac { 32 \sqrt { 65 } - 4 \sqrt { 5 } } { 3 } \pi$

Question 73

Evaluate the integral in cylindrical coordinates.

Question 74

The function has an average value of 16 on the rectangle with vertices at (0, 0),(0, 2), (2, 0)and (2, 2).Find the constant k.

Question 75

Let W be the region between the spheres $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$ and $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ .Given that $\int _ { W } \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 1 / 2 } d V = 15 \pi$ , evaluate the integral $\int _ { w } \left( 64 x ^ { 2 } + 36 y ^ { 2 } + 144 z ^ { 2 } \right) ^ { 1 / 2 } d V$ , where $\bar{W}$ is the region between the ellipsoids $\frac { x ^ { 2 } } { 3 ^ { 2 } } + \frac { y ^ { 2 } } { 4 ^ { 2 } } + \frac { z ^ { 2 } } { 2 ^ { 2 } } = 1$ and $\frac { x ^ { 2 } } { 3 ^ { 2 } } + \frac { y ^ { 2 } } { 4 ^ { 2 } } + \frac { z ^ { 2 } } { 2 ^ { 2 } } = 4$ .

A) $8640 \pi$
B) $2160 \pi$
C) $24 \pi$
D) $360 \pi$
E) $360 \pi ^ { 3 }$

Question 76

Find the condition on the non-negative constants a and b for p(x, y)to be a joint density function, where