Question 1

Evaluate the integral.&#10;

Accepted Answer

$$\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } ( 9 x - 8 y ) d y d x$$

First, we integrate with respect to $y$:

$$\int_{0}^{1}(9x-8y)dy$$

Using the power rule of integration, we have:

$$=9xy-4y^2 \Bigg|_{0}^{1}$$
$$=9x-4$$

Now we integrate with respect to $x$:

$$\int_{0}^{1}(9x-4)dx$$

Using the power rule, we have:

$$=\frac{9}{2}x^2-4x \Bigg|_{0}^{1}$$
$$=\frac{9}{2}-4$$
$$\boxed{=\frac{1}{2}}$$

Question 2

Evaluate the double integral over the given region.&#10;

Accepted Answer

The integral evaluates to $(\ln 3)^2$ because the integral of $1/(xy)$ over the region $3 \leq x \leq 9, 3 \leq y \leq 9$ results in $(\ln 3)^2$, which is equivalent to $\left( \ln \frac { 1 } { 3 } \right) ^ { 2 }$.

Question 3

Evaluate the integral.&#10;

Accepted Answer

First, integrate with respect to $y$: $\int_{-3}^{0} (8x + 5y) \, dy = [8xy + \frac{5}{2}y^2]_{-3}^{0} = 0 - (8x(-3) + \frac{5}{2}(-3)^2) = 24x - \frac{45}{2}$. Then, integrate with respect to $x$: $\int_{-10}^{0} (24x - \frac{45}{2}) \, dx = [12x^2 - \frac{45}{2}x]_{-10}^{0} = 0 - (12(-10)^2 - \frac{45}{2}(-10)) = -1425$.

Question 4

Evaluate the integral.&#10; &#10;A) 36450&#10;B) 4050&#10;C) 675&#10;D) 6075

Accepted Answer

Explanation:$$\int _ { 0 } ^ { 9 } \int _ { 0 } ^ { 6 } \left( 8 x ^ { 2 } y + 2 x y \right) d y d x$$$$= \int _ { 0 } ^ { 9 } \left [ 4 x ^ { 2 } y ^ { 2 } + x y ^ { 2 } \right ] _ { 0 } ^ { 6 } d x$$$$= \int _ { 0 } ^ { 9 } \left [ 4 x ^ { 2 } ( 6 ) ^ { 2 } + x ( 6 ) ^ { 2 } \right ] d x$$$$= \int _ { 0 } ^ { 9 } \left [ 144 x ^ { 2 } + 36 x \right ] d x$$$$= \left [ 48 x ^ { 3 } + 18 x ^ { 2 } \right ] _ { 0 } ^ { 9 }$$$$= 48 ( 9 ) ^ { 3 } + 18 ( 9 ) ^ { 2 }$$$$= 36450$$

Question 5

Evaluate the double integral over the given region.&#10;

Accepted Answer

The integral evaluates to $-\frac{43}{15}$ after calculating $\int_0^1 \int_0^2 (x + x^2 y^4 - 3) \, dy \, dx$.

Question 6

Evaluate the integral.&#10;

Accepted Answer

The integral can be separated into two parts because the integrand is a sum: $\int _ { 0 } ^ { 8 \pi } \int _ { 0 } ^ { 13 \pi } ( \sin x + \cos y ) d x d y = \int _ { 0 } ^ { 8 \pi } \int _ { 0 } ^ { 13 \pi } \sin x d x d y + \int _ { 0 } ^ { 8 \pi } \int _ { 0 } ^ { 13 \pi } \cos y d x d y$. The integral of $\sin x$ over one full period (0 to $2\pi$) is 0, so over $0$ to $13\pi$, it will also sum to 0 because $13\pi$ covers multiple full periods of sine. The integral of $\cos y$ over $0$ to $8\pi$ is also 0 for the same reason. However, the integral of $\cos y$ with respect to $x$ over the interval $0$ to $13\pi$ results in $13\pi \cos y$, and when integrated with respect to $y$ over $0$ to $8\pi$, the $\cos y$ part integrates to 0 over its full period. The mistake in this reasoning is that the integral of $\cos y$ with respect to $x$ over $0$ to $13\pi$ actually results in $13\pi \cos y$ being multiplied by the length of the interval in $x$, which is $13\pi$, and similarly for $\sin x$ over $y$. Correctly evaluating, the integral of $\sin x$ from $0$ to $13\pi$ does not contribute to the final answer due to its periodic nature, but the integral of $1$ with respect to $x$ over $0$ to $13\pi$ gives $13\pi$, which when multiplied by the integral of $\cos y$ over $0$ to $8\pi$, which is 0, leads to a misunderstanding of the problem's requirements. The correct approach is to recognize that the integral of a constant with respect to one variable over a finite interval results in the constant times the length of the interval. Therefore, the integral simplifies to evaluating the constant parts separately, leading to the correct solution without directly integrating $\sin x$ and $\cos y$ over their respective intervals. The correct calculation involves recognizing that the integral of $\sin x$ over $0$ to $13\pi$ and $\cos y$ over $0$ to $8\pi$ does not directly contribute to the solution in the explained manner, and a reevaluation of the integral's calculation is necessary to arrive at the correct answer.

Question 7

Evaluate the double integral over the given region.&#10;

Accepted Answer

The integral can be separated into two integrals due to the nature of the function: $\int_{0}^{6} \frac{1}{x+1} dx \cdot \int_{0}^{1} \frac{1}{y+1} dy$. Evaluating these separately gives $\ln(x+1)$ from 0 to 6 and $\ln(y+1)$ from 0 to 1, resulting in $\ln 7 \ln 2$.

Question 8

Evaluate the integral.&#10;

Accepted Answer

First, we need to integrate with respect to x and then with respect to y. 
$$\int _ { 0 } ^ { 3 } \int _ { 0 } ^ { 9 } ( 10 x + 5 y ) d x d y = \int _ { 0 } ^ { 9 } \int _ { 0 } ^ { 3 } ( 10 x + 5 y ) d x d y = \int _ { 0 } ^ { 9 } \left[ 5x^2+5yx \right] _ { x=0 } ^ { x=3 } dy$$
$$= \int _ { 0 } ^ { 9 } (45+15y) dy = \left[ 45y + \frac {15y^2}{2} \right] _ { y=0 } ^ { y=9 } = \frac {2835}{2}$$

Question 9

Evaluate the double integral over the given region.&#10;

Accepted Answer

Evaluate the integral by directly integrating over the given region:
$$\int_0^1\int_0^1 e^{2x+3y} dydx = \int_0^1 \left[\frac{e^{2x+3y}}{3}\right]_0^1 dx = \int_0^1 \frac{e^{2x+3}-e^{2x}}{3} dx = \left[\frac{e^{2x+3}}{6}-\frac{e^{2x}}{6}\right]_0^1=\frac{1}{6}(e^5-e^3-e^2+1)$$Therefore, the answer is choice A.

Question 10

Evaluate the integral.&#10; &#10;A) 14&#10;B) -25&#10;C) -5&#10;D) 1

Accepted Answer

The integral evaluates the area of a rectangle with sides of length $5 - (-2) = 7$ and $(-3) - (-5) = 2$, thus the area is $7 \times 2 = 14$.

Question 11

Integrate the function f over the given region.&#10;

Accepted Answer

The answer of Integrate the function f over the given...

Question 12

Evaluate the double integral over the given region.&#10;

Accepted Answer

The answer of Evaluate the double integral over the given...

Question 13

Evaluate the integral.&#10; &#10;A) 5&#10;B) 480&#10;C) 80&#10;D) 30

Accepted Answer

The answer of Evaluate the integral.&#10; &#10;A) 5&#10;B) 480&#10;C) 80&#10;D)...

Question 14

Evaluate the double integral over the given region.&#10;

Accepted Answer

The answer of Evaluate the double integral over the given...

Question 15

Evaluate the integral.&#10; &#10;A) 27&#10;B) 51&#10;C) 33&#10;D) 45

Accepted Answer

The answer of Evaluate the integral.&#10; &#10;A) 27&#10;B) 51&#10;C) 33&#10;D)...

Question 16

Evaluate the integral.&#10; &#10;A) - 10464&#10;B) 8736&#10;C) - 8736&#10;D) 10464

Accepted Answer

The answer of Evaluate the integral.&#10; &#10;A) - 10464&#10;B) 8736&#10;C)...

Question 17

Evaluate the double integral over the given region.&#10;

Accepted Answer

The answer of Evaluate the double integral over the given...

Question 18

Evaluate the double integral over the given region.&#10; &#10;A) 176&#10;B) 528&#10;C) 264&#10;D) 352

Accepted Answer

The answer of Evaluate the double integral over the given...

Question 19

Evaluate the integral.&#10; &#10;A) - 2730&#10;B) - 2535&#10;C) - 1170&#10;D) 1365

Accepted Answer

The answer of Evaluate the integral.&#10; &#10;A) - 2730&#10;B) -...

Question 20

Evaluate the integral.&#10; &#10;A) - 288&#10;B) - 1152&#10;C) - 2592&#10;D) 288

Accepted Answer

The answer of Evaluate the integral.&#10; &#10;A) - 288&#10;B) -...

Question 21

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 22

Integrate the function f over the given region.&#10;

Accepted Answer

The answer of Integrate the function f over the given...

Question 23

Find the volume under the surface z = f(x,y) and above the rectangle with the given boundaries.&#10; &#10;A) 576&#10;B) 1256&#10;C) 2256&#10;D) 676

Accepted Answer

The answer of Find the volume under the surface z...

Question 24

Evaluate the integral.&#10; &#10;A) 9&#10;B) 8&#10;C) 4&#10;D) 18

Accepted Answer

The answer of Evaluate the integral.&#10; &#10;A) 9&#10;B) 8&#10;C) 4&#10;D)...

Question 25

Integrate the function f over the given region.&#10;

Accepted Answer

The answer of Integrate the function f over the given...

Question 26

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 27

Integrate the function f over the given region.&#10;f(x, y) = xy over the triangular region with vertices (0, 0), (5, 0), and (0, 10)

Accepted Answer

The answer of Integrate the function f over the given...

Question 28

Integrate the function f over the given region.&#10;

Accepted Answer

The answer of Integrate the function f over the given...

Question 29

Integrate the function f over the given region.&#10;

Accepted Answer

The answer of Integrate the function f over the given...

Question 30

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 31

Find the volume under the surface z = f(x,y) and above the rectangle with the given boundaries.&#10; &#10;A) 36&#10;B) 26&#10;C) 28&#10;D) 38

Accepted Answer

The answer of Find the volume under the surface z...

Question 32

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 33

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 34

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 35

Find the volume under the surface z = f(x,y) and above the rectangle with the given boundaries.&#10;

Accepted Answer

The answer of Find the volume under the surface z...

Question 36

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 37

Find the volume under the surface z = f(x,y) and above the rectangle with the given boundaries.&#10;

Accepted Answer

The answer of Find the volume under the surface z...

Question 38

Integrate the function f over the given region.&#10;

Accepted Answer

The answer of Integrate the function f over the given...

Question 39

Find the volume under the surface z = f(x,y) and above the rectangle with the given boundaries.&#10;

Accepted Answer

The answer of Find the volume under the surface z...

Question 40

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 41

Write an equivalent double integral with the order of integration reversed.&#10;

Accepted Answer

The answer of Write an equivalent double integral with the...

Question 42

Integrate the function f over the given region.&#10;

Accepted Answer

The answer of Integrate the function f over the given...

Question 43

Write an equivalent double integral with the order of integration reversed.&#10;

Accepted Answer

The answer of Write an equivalent double integral with the...

Question 44

Write an equivalent double integral with the order of integration reversed.&#10;

Accepted Answer

The answer of Write an equivalent double integral with the...

Question 45

Write an equivalent double integral with the order of integration reversed.&#10;

Accepted Answer

The answer of Write an equivalent double integral with the...

Question 46

Write an equivalent double integral with the order of integration reversed.&#10;

Accepted Answer

The answer of Write an equivalent double integral with the...

Question 47

Reverse the order of integration and then evaluate the integral.&#10; &#10;A) 9 cos 7&#10;B) 9 sin 7&#10;C) 7 sin 9&#10;D) 7 cos 9

Accepted Answer

The answer of Reverse the order of integration and then...

Question 48

Write an equivalent double integral with the order of integration reversed.&#10;

Accepted Answer

The answer of Write an equivalent double integral with the...

Question 49

Write an equivalent double integral with the order of integration reversed.&#10;

Accepted Answer

The answer of Write an equivalent double integral with the...

Question 50

Reverse the order of integration and then evaluate the integral.&#10; &#10;A) 1 + cos 5&#10;B) cos 5&#10;C) - cos 5&#10;D) 1 - cos 5

Accepted Answer

The answer of Reverse the order of integration and then...

Question 51

Integrate the function f over the given region.&#10;

Accepted Answer

The answer of Integrate the function f over the given...

Question 52

Write an equivalent double integral with the order of integration reversed.&#10;

Accepted Answer

The answer of Write an equivalent double integral with the...

Question 53

Reverse the order of integration and then evaluate the integral.&#10;

Accepted Answer

The answer of Reverse the order of integration and then...

Question 54

Reverse the order of integration and then evaluate the integral.&#10;

Accepted Answer

The answer of Reverse the order of integration and then...

Question 55

Write an equivalent double integral with the order of integration reversed.&#10;

Accepted Answer

The answer of Write an equivalent double integral with the...

Question 56

Reverse the order of integration and then evaluate the integral.&#10;

Accepted Answer

The answer of Reverse the order of integration and then...

Question 57

Reverse the order of integration and then evaluate the integral.&#10;

Accepted Answer

The answer of Reverse the order of integration and then...

Question 58

Write an equivalent double integral with the order of integration reversed.&#10;

Accepted Answer

The answer of Write an equivalent double integral with the...

Question 59

Reverse the order of integration and then evaluate the integral.&#10; &#10;A) 25&#10;B) 35&#10;C) 33&#10;D) 27

Accepted Answer

The answer of Reverse the order of integration and then...

Question 60

Integrate the function f over the given region.&#10; &#10;A) 6&#10;B) 1&#10;C) 4&#10;D) 5

Accepted Answer

The answer of Integrate the function f over the given...

Question 61

Find the volume of the indicated region.&#10;

Accepted Answer

The answer of Find the volume of the indicated region.&#10;...

Question 62

Find the volume of the indicated region.&#10;

Accepted Answer

The answer of Find the volume of the indicated region.&#10;...

Question 63

Evaluate the improper integral.&#10;

Accepted Answer

The answer of Evaluate the improper integral.&#10; ...

Question 64

Evaluate the improper integral.&#10;

Accepted Answer

The answer of Evaluate the improper integral.&#10; ...

Deck 16: Multiple Integrals