Question 1

Evaluate the following iterated integral. $\int _ { 2 } ^ { 5 } \int _ { 4 } ^ { 5 } ( 2 x + y ) d y d x$

Accepted Answer

A)  $41.5$ 
B)  $38.5$ 
C)  $34.5$ 
D)  $33.5$ 
E)  $37.5$ 
A)  $41.5$ 
B)  $38.5$ 
C)  $34.5$ 
D)  $33.5$ 
E)  $37.5$

Question 2

$\text { Evaluate } \int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 1 } ( 1 + 2 x + 2 y ) d y d x$

Accepted Answer

A) 3 
B) 8 
C) 6 
D) 9 
E) 5 
A) 3 
B) 8 
C) 6 
D) 9 
E) 5

Question 3

$\text { Evaluate the iterated integral } \int _ { 0 } ^ { 2 } \int _ { y ^ { 2 } } ^ { 4 } \sqrt { x } \sin ( 2 x ) d x d y \text { by switching the order of }$ integration. Round your answer to three decimal places.

Accepted Answer

A)  $0.538$ 
B)  $0.558$ 
C)  $30.538$ 
D)  $1.538$ 
E)  $4.538$ 
A)  $0.538$ 
B)  $0.558$ 
C)  $30.538$ 
D)  $1.538$ 
E)  $4.538$

Question 4

Evaluate the following iterated integral. $\int _ { 5 } ^ { 8 } \int _ { 1 } ^ { \sqrt { x } } 2 y e ^ { - x } d y d x$

Accepted Answer

A)  $5 e ^ { 5 } - 8 e ^ { 8 }$ 
B)  $5 e ^ { - 5 } - 8 e ^ { - 8 }$ 
C)  $5 e ^ { - \sqrt { 5 } } - 8 e ^ { - \sqrt { 8 } }$ 
D)  $5 e ^ { - 5 } + 8 e ^ { - 8 }$ 
E)  $8 e ^ { - 8 } - 5 e ^ { - 5 }$ 
A)  $5 e ^ { 5 } - 8 e ^ { 8 }$ 
B)  $5 e ^ { - 5 } - 8 e ^ { - 8 }$ 
C)  $5 e ^ { - \sqrt { 5 } } - 8 e ^ { - \sqrt { 8 } }$ 
D)  $5 e ^ { - 5 } + 8 e ^ { - 8 }$ 
E)  $8 e ^ { - 8 } - 5 e ^ { - 5 }$

Question 5

Evaluate the following integral. $$\int _ { 11 } ^ { 5 y } \frac { - 4 y } { x } d x$$

Accepted Answer

A)  $- 4 \ln \left( \frac { 5 y } { 11 } \right)$ 
B)  $- 4 ( \ln ( 5 y ) - \ln ( 11 ) )$ 
C)  $- 4 y ( \ln ( 5 y ) + \ln ( 11 ) )$ 
D)  $y \ln \left( \frac { 5 y } { 11 } \right)$ 
E) $- 4 y \ln \left( \frac { 5 y } { 11 } \right)$ 
A)  $- 4 \ln \left( \frac { 5 y } { 11 } \right)$ 
B)  $- 4 ( \ln ( 5 y ) - \ln ( 11 ) )$ 
C)  $- 4 y ( \ln ( 5 y ) + \ln ( 11 ) )$ 
D)  $y \ln \left( \frac { 5 y } { 11 } \right)$ 
E) $- 4 y \ln \left( \frac { 5 y } { 11 } \right)$

Question 6

Find the center of mass of the lamina bounded by the graphs of the equations $y = e ^ { x } , y = 0 , x = 0$, and $x = 20$ for the density $\rho = k$

Accepted Answer

A)  $\left( \frac { 1 - 21 e ^ { 21 } } { e ^ { 20 } - 1 } , \frac { e ^ { 21 } + 1 } { 2 } \right)$ 
B)  $\left( \frac { 1 + 19 e ^ { 20 } } { e ^ { 20 } - 1 } , \frac { e ^ { 20 } + 1 } { 4 } \right)$ 
C)  $\left( \frac { 1 - 19 e ^ { 20 } } { e ^ { 20 } + 1 } , \frac { e ^ { 20 } - 1 } { 4 } \right)$ 
D) $\left( \frac { 1 + 21 e ^ { 21 } } { e ^ { 20 } - 1 } , \frac { e ^ { 21 } - 1 } { 4 } \right)$ 
E)  $\left( \frac { 1 + 19 e ^ { 20 } } { e ^ { 20 } + 1 } , \frac { e ^ { 20 } + 1 } { 2 } \right)$ 
A)  $\left( \frac { 1 - 21 e ^ { 21 } } { e ^ { 20 } - 1 } , \frac { e ^ { 21 } + 1 } { 2 } \right)$ 
B)  $\left( \frac { 1 + 19 e ^ { 20 } } { e ^ { 20 } - 1 } , \frac { e ^ { 20 } + 1 } { 4 } \right)$ 
C)  $\left( \frac { 1 - 19 e ^ { 20 } } { e ^ { 20 } + 1 } , \frac { e ^ { 20 } - 1 } { 4 } \right)$ 
D) $\left( \frac { 1 + 21 e ^ { 21 } } { e ^ { 20 } - 1 } , \frac { e ^ { 21 } - 1 } { 4 } \right)$ 
E)  $\left( \frac { 1 + 19 e ^ { 20 } } { e ^ { 20 } + 1 } , \frac { e ^ { 20 } + 1 } { 2 } \right)$

Question 7

A company produces a spherical object of radius 17 centimeters. A hole of radius 7 centimeters is drilled through the center of the object. Find the volume of the object.

Accepted Answer

A)  $1,344 \pi \sqrt { 15 } \mathrm {~cm} ^ { 3 }$ 
B)  $1,680 \pi \sqrt { 15 } \mathrm {~cm} ^ { 3 }$ 
C)  $1,280 \pi \sqrt { 15 } \mathrm {~cm} ^ { 3 }$ 
D)  $3,360 \pi \sqrt { 15 } \mathrm {~cm} ^ { 3 }$ 
E)  $1,120 \pi \sqrt { 15 } \mathrm {~cm} ^ { 3 }$ 
A)  $1,344 \pi \sqrt { 15 } \mathrm {~cm} ^ { 3 }$ 
B)  $1,680 \pi \sqrt { 15 } \mathrm {~cm} ^ { 3 }$ 
C)  $1,280 \pi \sqrt { 15 } \mathrm {~cm} ^ { 3 }$ 
D)  $3,360 \pi \sqrt { 15 } \mathrm {~cm} ^ { 3 }$ 
E)  $1,120 \pi \sqrt { 15 } \mathrm {~cm} ^ { 3 }$

Question 8

Use a triple integral to find the volume of the solid bounded by the graphs of the equations $z = 2 - y , z = 6 - y ^ { 2 } , x = 0 , x = 3 , y = 0$

Accepted Answer

A)  22 
B)  32 
C)  34 
D)  26 
E)  38 
A)  22 
B)  32 
C)  34 
D)  26 
E)  38

Question 9

Find the area of the surface given by $z = f ( x , y ) \text { over the region } R \text {. }$ $\begin{array} { l } 
f ( x , y ) = x y \
R = \left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } \leq 100 \right\}
\end{array}$

Accepted Answer

A)  $\frac { 2 } { 3 } ( 101 \sqrt { 101 } - 1 ) \pi$ 
B)  $\frac { 2 } { 3 } ( 1 - 101 \sqrt { 101 } ) \pi$ 
C)  $\frac { 2 } { 3 } ( 101 \sqrt { 101 } - 1 )$ 
D)  $\frac { 3 } { 4 } ( 101 \sqrt { 101 } - 1 ) \pi$ 
E)  $\frac { 2 } { 3 } ( 100 \sqrt { 101 } - 1 ) \pi$ 
A)  $\frac { 2 } { 3 } ( 101 \sqrt { 101 } - 1 ) \pi$ 
B)  $\frac { 2 } { 3 } ( 1 - 101 \sqrt { 101 } ) \pi$ 
C)  $\frac { 2 } { 3 } ( 101 \sqrt { 101 } - 1 )$ 
D)  $\frac { 3 } { 4 } ( 101 \sqrt { 101 } - 1 ) \pi$ 
E)  $\frac { 2 } { 3 } ( 100 \sqrt { 101 } - 1 ) \pi$

Question 10

Use a double integral in polar coordinates to find the volume of the solid inside the hemisphere $z = \sqrt { 64 - x ^ { 2 } - y ^ { 2 } }$ but outside the cylinder $x ^ { 2 } + y ^ { 2 } = 25$

Accepted Answer

A)  $\frac { 2 } { 3 } 39 ^ { 3 / 2 }$ 
B)  $\frac { 2 } { 3 } 39 ^ { 3 / 2 } \pi$ 
C)  $\frac { 4 } { 3 } 39 ^ { 3 / 2 } \pi$ 
D)  $\frac { 2 } { 3 } 39 ^ { 1 / 2 } \pi$ 
E)  $\frac { 3 } { 2 } 39 ^ { 3 / 2 }$ 
A)  $\frac { 2 } { 3 } 39 ^ { 3 / 2 }$ 
B)  $\frac { 2 } { 3 } 39 ^ { 3 / 2 } \pi$ 
C)  $\frac { 4 } { 3 } 39 ^ { 3 / 2 } \pi$ 
D)  $\frac { 2 } { 3 } 39 ^ { 1 / 2 } \pi$ 
E)  $\frac { 3 } { 2 } 39 ^ { 3 / 2 }$

Question 11

Use a change of variables to find the volume of the solid region lying below the surface $z = 27 x y$ and above the plane region $R$ : region bounded by the square with vertices $( 0,0 ) , ( - 2,2 ) , ( 0,4 ) , ( 2,2 )$

Accepted Answer

A)  432 
B)  72 
C)  36 
D)  0 
E)  108 
A)  432 
B)  72 
C)  36 
D)  0 
E)  108

Question 12

Evaluate the following iterated integral. 
$\int_{0}^{6}\int_{0} ^ {4}\int_{0} ^ {2}(-4 x+5 y+z) d x d y d z $

Accepted Answer

A)  428 
B)  438 
C)  431 
D)  442 
E)  432 
A)  428 
B)  438 
C)  431 
D)  442 
E)  432

Question 13

$\text { Write a double integral that represents the surface area of } f ( x , y ) = 12 - x ^ { 2 } - y ^ { 2 }$ over the region $R : R = \{ ( x , y ) : 0 \leq x \leq 4,0 \leq y \leq 4 \}$. Use a computer algebra system to evaluate the double integral. Round your answer to four decimal places.

Accepted Answer

A)  $4.5062$ 
B)  $140.1398$ 
C)  $37.8613$ 
D)  $99.5824$ 
E)  $13.6778$ 
A)  $4.5062$ 
B)  $140.1398$ 
C)  $37.8613$ 
D)  $99.5824$ 
E)  $13.6778$

Question 14

Find the mass of the lamina described by the inequalities $x \geq 0$ and $3 \leq y \leq 3 + \sqrt { 9 - x ^ { 2 } }$, given that its density is $\rho ( x , y ) = 7 x y$. (Hint: Some of the integrals are simpler in polar coordinates.)

Accepted Answer

A)  $\frac { 2,079 } { 8 }$ 
B)  $\frac { 16 } { 6,237 }$ 
C)  $\frac { 8 } { 2,079 }$ 
D)  $\frac { 2,457 } { 8 }$ 
E)  $\frac { 6,237 } { 16 }$ 
A)  $\frac { 2,079 } { 8 }$ 
B)  $\frac { 16 } { 6,237 }$ 
C)  $\frac { 8 } { 2,079 }$ 
D)  $\frac { 2,457 } { 8 }$ 
E)  $\frac { 6,237 } { 16 }$

Question 15

Rewrite the iterated integral $ \int_{0}^{6} \int_{0}^{\frac{6-x}{2}}\int_{0}^{\frac{24-4x-8y}{8}} dzdydx $ using the order $d y d x d z$.

Accepted Answer

A) 
$ \int_{0}^{24} \int_{0}^{\frac{24-x}{8}}\int_{0}^{\frac{24-3x-x}{4}} dydxdz $ 
B) 
$ \int_{0}^{8} \int_{0}^{\frac{24-3x}{3}}\int_{0}^{\frac{24-3x-4x}{8}} dydxdz $ 
C)  
$ \int_{0}^{8} \int_{0}^{\frac{24-3x}{3}}\int_{0}^{\frac{24-3x-x}{4}} dydxdz $ 
D) 
$ \int_{0}^{6} \int_{0}^{\frac{24-x}{3}}\int_{0}^{\frac{24-3x-4x}{8}} dydxdz $ 
E) 
$ \int_{0}^{24} \int_{0}^{\frac{24-3x}{4}}\int_{0}^{\frac{24-3x-4x}{8}} dydxdz $ 
A) 
$ \int_{0}^{24} \int_{0}^{\frac{24-x}{8}}\int_{0}^{\frac{24-3x-x}{4}} dydxdz $ 
B) 
$ \int_{0}^{8} \int_{0}^{\frac{24-3x}{3}}\int_{0}^{\frac{24-3x-4x}{8}} dydxdz $ 
C)  
$ \int_{0}^{8} \int_{0}^{\frac{24-3x}{3}}\int_{0}^{\frac{24-3x-x}{4}} dydxdz $ 
D) 
$ \int_{0}^{6} \int_{0}^{\frac{24-x}{3}}\int_{0}^{\frac{24-3x-4x}{8}} dydxdz $ 
E) 
$ \int_{0}^{24} \int_{0}^{\frac{24-3x}{4}}\int_{0}^{\frac{24-3x-4x}{8}} dydxdz $

Question 16

Determine the diameter of a hole that is drilled vertically through the center of the solid bounded by the graphs of the equations $z = 28 e ^ { - \left( x ^ { 2 } + y ^ { 2 } \right) / 4 } , z = 0$, and $x ^ { 2 } + y ^ { 2 } = 9$ if one-tenth of the volume of the solid is removed. Round your answer to four decimal places.

Accepted Answer

A)  $1.2245$ 
B)  $31.4490$ 
C)  $7.2245$ 
D)  $5.4490$ 
E)  $15.2245$ 
A)  $1.2245$ 
B)  $31.4490$ 
C)  $7.2245$ 
D)  $5.4490$ 
E)  $15.2245$

Question 17

Suppose the population density of a city is approximated by the model $f ( x , y ) = 6000 e ^ { - 0.01 \left( x ^ { 2 } + y ^ { 2 } \right) } , x ^ { 2 } + y ^ { 2 } \leq 25$, where $x$ and $y$ are measured in miles. Integrate the density function over the indicated circular region to approximate the population of the city. Round your answer to the nearest integer.

Accepted Answer

A)  417,127 
B)  417,029 
C)  833,901 
D)  833,903 
E)  416,951 
A)  417,127 
B)  417,029 
C)  833,901 
D)  833,903 
E)  416,951

Question 18

Evaluate the following iterated integral. $$\int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { \cos ^ { 2 } \theta } \int _ { 0 } ^ { 1 - r ^ { 2 } } r \sin \theta d z d r d \theta$$

Accepted Answer

A)  $\frac { 11 } { 180 }$ 
B)  $\frac { 13 } { 180 }$ 
C)  $\frac { 1 } { 12 }$ 
D)  $\frac { 180 } { 11 }$ 
E)  $\frac { 12 } { 1 }$ 
A)  $\frac { 11 } { 180 }$ 
B)  $\frac { 13 } { 180 }$ 
C)  $\frac { 1 } { 12 }$ 
D)  $\frac { 180 } { 11 }$ 
E)  $\frac { 12 } { 1 }$

Question 19

Use an iterated integral to find the area of the region shown in the figure below.

Accepted Answer

A)  $\frac { 1024 } { 3 }$ 
B)  $\frac { 16 } { 3 }$ 
C)  $\frac { 125 } { 3 }$ 
D)  $\frac { 17 } { 3 }$ 
E)  $\frac { 128 } { 3 }$ 
A)  $\frac { 1024 } { 3 }$ 
B)  $\frac { 16 } { 3 }$ 
C)  $\frac { 125 } { 3 }$ 
D)  $\frac { 17 } { 3 }$ 
E)  $\frac { 128 } { 3 }$

Question 20

up a triple integral for the volume of the solid bounded above by the cylinder $z = 14 - x ^ { 2 }$ and below by the paraboloid $z = x ^ { 2 } + 15 y ^ { 2 }$.

Accepted Answer

A) 
$ \int_{-\sqrt{7}}^{\sqrt{7}} \int_{-\sqrt{\frac{b-x^2}{a}}}^{\sqrt{\frac{b-x^2}{a}}} \int_{x^2+15y^2}^{14-x^2}  d z dy d x $ 
B) 
$ \int_{-\sqrt{14}}^{\sqrt{7}} \int_{-\sqrt{\frac{b-x^2}{a}}}^{\sqrt{\frac{b-x^2}{a}}} \int_{14-x^2}^{x^2+15y^2}  d z dy d x $ 
C)  
$ \int_{-\sqrt{14}}^{\sqrt{14}} \int_{-\sqrt{\frac{b-x^2}{a}}}^{\sqrt{\frac{b-x^2}{a}}} \int_{x^2+15y^2}^{14-x^2} = d z dx d y $ 
D) 
$ \int_{-\sqrt{7}}^{\sqrt{7}} \int_{-\sqrt{\frac{14-2x^2}{a}}}^{\sqrt{\frac{14-2x^2}{a}}} \int_{x^2+15y^2}^{14-x^2} = d z dy d x $ 
E) 
$ \int_{-\sqrt{7}}^{\sqrt{14}} \int_{-\sqrt{\frac{b-x^2}{a}}}^{\sqrt{\frac{b-x^2}{a}}} \int_{14-x^2}^{x^2+15y^2} = d z dy d x $ 
A) 
$ \int_{-\sqrt{7}}^{\sqrt{7}} \int_{-\sqrt{\frac{b-x^2}{a}}}^{\sqrt{\frac{b-x^2}{a}}} \int_{x^2+15y^2}^{14-x^2}  d z dy d x $ 
B) 
$ \int_{-\sqrt{14}}^{\sqrt{7}} \int_{-\sqrt{\frac{b-x^2}{a}}}^{\sqrt{\frac{b-x^2}{a}}} \int_{14-x^2}^{x^2+15y^2}  d z dy d x $ 
C)  
$ \int_{-\sqrt{14}}^{\sqrt{14}} \int_{-\sqrt{\frac{b-x^2}{a}}}^{\sqrt{\frac{b-x^2}{a}}} \int_{x^2+15y^2}^{14-x^2} = d z dx d y $ 
D) 
$ \int_{-\sqrt{7}}^{\sqrt{7}} \int_{-\sqrt{\frac{14-2x^2}{a}}}^{\sqrt{\frac{14-2x^2}{a}}} \int_{x^2+15y^2}^{14-x^2} = d z dy d x $ 
E) 
$ \int_{-\sqrt{7}}^{\sqrt{14}} \int_{-\sqrt{\frac{b-x^2}{a}}}^{\sqrt{\frac{b-x^2}{a}}} \int_{14-x^2}^{x^2+15y^2} = d z dy d x $

Evaluate the following iterated integral. $\int _ { 2 } ^ { 5 } \int _ { 4 } ^ { 5 } ( 2 x + y ) d y d x$

$\text { Evaluate } \int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 1 } ( 1 + 2 x + 2 y ) d y d x$

$\text { Evaluate the iterated integral } \int _ { 0 } ^ { 2 } \int _ { y ^ { 2 } } ^ { 4 } \sqrt { x } \sin ( 2 x ) d x d y \text { by switching the order of }$ integration. Round your answer to three decimal places.

Evaluate the following iterated integral. $\int _ { 5 } ^ { 8 } \int _ { 1 } ^ { \sqrt { x } } 2 y e ^ { - x } d y d x$

Evaluate the following integral. $\int _ { 11 } ^ { 5 y } \frac { - 4 y } { x } d x$

Find the center of mass of the lamina bounded by the graphs of the equations $y = e ^ { x } , y = 0 , x = 0$ , and $x = 20$ for the density $\rho = k$

A company produces a spherical object of radius 17 centimeters. A hole of radius 7 centimeters is drilled through the center of the object. Find the volume of the object.

Use a triple integral to find the volume of the solid bounded by the graphs of the equations $z = 2 - y , z = 6 - y ^ { 2 } , x = 0 , x = 3 , y = 0$

Find the area of the surface given by $z = f ( x , y ) \text { over the region } R \text {. }$ f(x,y)=xy R= (x,y):+\leq100

Use a double integral in polar coordinates to find the volume of the solid inside the hemisphere $z = \sqrt { 64 - x ^ { 2 } - y ^ { 2 } }$ but outside the cylinder $x ^ { 2 } + y ^ { 2 } = 25$

Use a change of variables to find the volume of the solid region lying below the surface $z = 27 x y$ and above the plane region $R$ : region bounded by the square with vertices $( 0,0 ) , ( - 2,2 ) , ( 0,4 ) , ( 2,2 )$

Evaluate the following iterated integral. $\int_{0}^{6}\int_{0} ^ {4}\int_{0} ^ {2}(-4 x+5 y+z) d x d y d z$

$\text { Write a double integral that represents the surface area of } f ( x , y ) = 12 - x ^ { 2 } - y ^ { 2 }$ over the region $R : R = \{ ( x , y ) : 0 \leq x \leq 4,0 \leq y \leq 4 \}$ . Use a computer algebra system to evaluate the double integral. Round your answer to four decimal places.

Find the mass of the lamina described by the inequalities $x \geq 0$ and $3 \leq y \leq 3 + \sqrt { 9 - x ^ { 2 } }$ , given that its density is $\rho ( x , y ) = 7 x y$ . (Hint: Some of the integrals are simpler in polar coordinates.)

Rewrite the iterated integral $\int_{0}^{6} \int_{0}^{\frac{6-x}{2}}\int_{0}^{\frac{24-4x-8y}{8}} dzdydx$ using the order $d y d x d z$ .

Evaluate the following iterated integral. $\int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { \cos ^ { 2 } \theta } \int _ { 0 } ^ { 1 - r ^ { 2 } } r \sin \theta d z d r d \theta$

Use an iterated integral to find the area of the region shown in the figure below.

up a triple integral for the volume of the solid bounded above by the cylinder $z = 14 - x ^ { 2 }$ and below by the paraboloid $z = x ^ { 2 } + 15 y ^ { 2 }$ .

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Vector Fields

Exact First-Order Equations

Filters

Exam 14: Iterated Integrals and Area in the Plane

Evaluate the following iterated integral. ∫25∫45(2x+y)dydx\int _ { 2 } ^ { 5 } \int _ { 4 } ^ { 5 } ( 2 x + y ) d y d x∫25​∫45​(2x+y)dydx

Evaluate ∫02∫01(1+2x+2y)dydx\text { Evaluate } \int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 1 } ( 1 + 2 x + 2 y ) d y d x Evaluate ∫02​∫01​(1+2x+2y)dydx

Evaluate the following iterated integral. ∫58∫1x2ye−xdydx\int _ { 5 } ^ { 8 } \int _ { 1 } ^ { \sqrt { x } } 2 y e ^ { - x } d y d x∫58​∫1x​​2ye−xdydx

Evaluate the following integral. ∫115y−4yxdx\int _ { 11 } ^ { 5 y } \frac { - 4 y } { x } d x∫115y​x−4y​dx

Find the center of mass of the lamina bounded by the graphs of the equations y=ex,y=0,x=0y = e ^ { x } , y = 0 , x = 0y=ex,y=0,x=0 , and x=20x = 20x=20 for the density ρ=k\rho = kρ=k

A company produces a spherical object of radius 17 centimeters. A hole of radius 7 centimeters is drilled through the center of the object. Find the volume of the object.

Use a triple integral to find the volume of the solid bounded by the graphs of the equations z=2−y,z=6−y2,x=0,x=3,y=0z = 2 - y , z = 6 - y ^ { 2 } , x = 0 , x = 3 , y = 0z=2−y,z=6−y2,x=0,x=3,y=0

Find the area of the surface given by z=f(x,y) over the region R. z = f ( x , y ) \text { over the region } R \text {. }z=f(x,y) over the region R. f(x,y)=xy R= (x,y):+\leq100

Use a double integral in polar coordinates to find the volume of the solid inside the hemisphere z=64−x2−y2z = \sqrt { 64 - x ^ { 2 } - y ^ { 2 } }z=64−x2−y2​ but outside the cylinder x2+y2=25x ^ { 2 } + y ^ { 2 } = 25x2+y2=25

Use a change of variables to find the volume of the solid region lying below the surface z=27xyz = 27 x yz=27xy and above the plane region RRR : region bounded by the square with vertices (0,0),(−2,2),(0,4),(2,2)( 0,0 ) , ( - 2,2 ) , ( 0,4 ) , ( 2,2 )(0,0),(−2,2),(0,4),(2,2)

Evaluate the following iterated integral. ∫06∫04∫02(−4x+5y+z)dxdydz\int_{0}^{6}\int_{0} ^ {4}\int_{0} ^ {2}(-4 x+5 y+z) d x d y d z ∫06​∫04​∫02​(−4x+5y+z)dxdydz

Find the mass of the lamina described by the inequalities x≥0x \geq 0x≥0 and 3≤y≤3+9−x23 \leq y \leq 3 + \sqrt { 9 - x ^ { 2 } }3≤y≤3+9−x2​ , given that its density is ρ(x,y)=7xy\rho ( x , y ) = 7 x yρ(x,y)=7xy . (Hint: Some of the integrals are simpler in polar coordinates.)

Rewrite the iterated integral ∫06∫06−x2∫024−4x−8y8dzdydx \int_{0}^{6} \int_{0}^{\frac{6-x}{2}}\int_{0}^{\frac{24-4x-8y}{8}} dzdydx ∫06​∫026−x​​∫0824−4x−8y​​dzdydx using the order dydxdzd y d x d zdydxdz .

Evaluate the following iterated integral. ∫0π/2∫0cos⁡2θ∫01−r2rsin⁡θdzdrdθ\int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { \cos ^ { 2 } \theta } \int _ { 0 } ^ { 1 - r ^ { 2 } } r \sin \theta d z d r d \theta∫0π/2​∫0cos2θ​∫01−r2​rsinθdzdrdθ

Use an iterated integral to find the area of the region shown in the figure below.

up a triple integral for the volume of the solid bounded above by the cylinder z=14−x2z = 14 - x ^ { 2 }z=14−x2 and below by the paraboloid z=x2+15y2z = x ^ { 2 } + 15 y ^ { 2 }z=x2+15y2 .

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Vector Fields

Exact First-Order Equations

Filters

Evaluate the following iterated integral. $\int _ { 2 } ^ { 5 } \int _ { 4 } ^ { 5 } ( 2 x + y ) d y d x$

$\text { Evaluate } \int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 1 } ( 1 + 2 x + 2 y ) d y d x$

Evaluate the following iterated integral. $\int _ { 5 } ^ { 8 } \int _ { 1 } ^ { \sqrt { x } } 2 y e ^ { - x } d y d x$

Evaluate the following integral. $\int _ { 11 } ^ { 5 y } \frac { - 4 y } { x } d x$

Find the center of mass of the lamina bounded by the graphs of the equations $y = e ^ { x } , y = 0 , x = 0$ , and $x = 20$ for the density $\rho = k$

Use a triple integral to find the volume of the solid bounded by the graphs of the equations $z = 2 - y , z = 6 - y ^ { 2 } , x = 0 , x = 3 , y = 0$

Find the area of the surface given by $z = f ( x , y ) \text { over the region } R \text {. }$ f(x,y)=xy R= (x,y):+\leq100

Use a double integral in polar coordinates to find the volume of the solid inside the hemisphere $z = \sqrt { 64 - x ^ { 2 } - y ^ { 2 } }$ but outside the cylinder $x ^ { 2 } + y ^ { 2 } = 25$

Use a change of variables to find the volume of the solid region lying below the surface $z = 27 x y$ and above the plane region $R$ : region bounded by the square with vertices $( 0,0 ) , ( - 2,2 ) , ( 0,4 ) , ( 2,2 )$

Evaluate the following iterated integral. $\int_{0}^{6}\int_{0} ^ {4}\int_{0} ^ {2}(-4 x+5 y+z) d x d y d z$

Find the mass of the lamina described by the inequalities $x \geq 0$ and $3 \leq y \leq 3 + \sqrt { 9 - x ^ { 2 } }$ , given that its density is $\rho ( x , y ) = 7 x y$ . (Hint: Some of the integrals are simpler in polar coordinates.)

Rewrite the iterated integral $\int_{0}^{6} \int_{0}^{\frac{6-x}{2}}\int_{0}^{\frac{24-4x-8y}{8}} dzdydx$ using the order $d y d x d z$ .

Evaluate the following iterated integral. $\int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { \cos ^ { 2 } \theta } \int _ { 0 } ^ { 1 - r ^ { 2 } } r \sin \theta d z d r d \theta$

up a triple integral for the volume of the solid bounded above by the cylinder $z = 14 - x ^ { 2 }$ and below by the paraboloid $z = x ^ { 2 } + 15 y ^ { 2 }$ .