Question 1

Evaluate the integral $\int _ { 0 } ^ { 3 } \int _ { - y / 3 } ^ { 0 } e ^ { y ^ { 2 } } d x d y$ .Give your answer to two decimal places.

Accepted Answer

The answer of Evaluate the integral \(\int _ { 0...

Question 2

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 x > y +0.4.

Accepted Answer

The answer of Let x and y have joint density...

Question 3

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.

Accepted Answer

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)  $\text { Mass } = \int _ { 0 } ^ { 1 } \int _ { - \sqrt { 1 - x ^ { 2 } } } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { x ^ { 2 } + y ^ { 2 } } ^ { 1 } z ^ { 2 } d z d y d x$ 
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 4

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.

Accepted Answer

The answer of A solid is bounded below by the...

Question 5

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.

Accepted Answer

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)  $\int _ { R } y d V = \int _ { - 1 } ^ { 1 } \int _ { - \sqrt { 1 - x ^ { 2 } } } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { \sqrt { 3 x ^ { 2 } + 3 y ^ { 2 } } } ^ { \sqrt { 4 - x ^ { 2 } - y ^ { 2 } } } y d z d y d x$ 
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 6

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 $\int _ { R } \left( 4 x ^ { 2 } + 9 y ^ { 2 } \right) d A$

Accepted Answer

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

Question 7

Find the mass of the solid cylinder $x ^ { 2 } + y ^ { 2 } \leq 25$ , $4 \leq Z \leq 6$ with density function $f ( x , y , z ) = z + x ^ { 2 } + y ^ { 2 }$

Accepted Answer

The answer of Find the mass of the solid cylinder...

Question 8

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

Accepted Answer

The answer of Compute the area of the flower-like region...

Question 9

The joint density function for random variables x and y is $f ( x , y ) = \left\{ \begin{array} { c c } 
\frac { 1 } { 21 } ( x + y ) & \text { if } 0 \leq x \leq 3,0 \leq y \leq 2 \
0 & \text { otherwise }
\end{array} \right.$ Find the probability $P ( x \leq 2.5 , y \geq 0.2 )$ .Give your answer to 3 decimal places.

Accepted Answer

The answer of The joint density function for random variables...

Question 10

Upper and lower sums for a function f on a rectangle R, using n subdivisions on each side, are $\left( 3 n ^ { 2 } + 8 n + 24 \right) / n ^ { 2 }$ and $\left( 3 n ^ { 2 } + 7 n + 8 \right) / n ^ { 2 }$ respectively.Evaluate $\int _ { R } f d A$

Accepted Answer

The answer of Upper and lower sums for a function...

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.

Accepted Answer

The answer of Let f(x, y)be a positive function of...

Question 12

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.

Accepted Answer

The answer of Consider the volume between a cone centered...

Question 13

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

Accepted Answer

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)  $\int _ { \pi / 4 } ^ { 3 \pi / 4 } \int _ { 1 / \sin \theta } ^ { \sqrt { 2 } } f ( r \cos \theta , r \sin \theta ) r d \theta d r$ 
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 14

The function $f ( x , y ) = c x ^ { 2 } + 5 y$ has an average value of 4 on the triangle with vertices at (0, 0), (0, 1)and (1, 0).Find the constant a.

Accepted Answer

The answer of The function \(f ( x , y...

Question 15

The function $f ( x ) = k x ^ { 2 } + 4 y$ 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.

Accepted Answer

The answer of The function \(f ( x ) =...

Question 16

Evaluate the integral $\int _ { 4 } ^ { 6 } \int _ { - 1 } ^ { - 1 } \int _ { - \sqrt { 16 - y ^ { 2 } } } ^ { \sqrt { 16 - y ^ { 2 } } } \frac { 1 } { \sqrt { x ^ { 2 } + y ^ { 2 } } } d x d y d z$ in cylindrical coordinates.

Accepted Answer

The answer of Evaluate the integral \(\int _ { 4...

Evaluate the integral $\int _ { 0 } ^ { 3 } \int _ { - y / 3 } ^ { 0 } e ^ { y ^ { 2 } } d x d y$ .Give your answer to two decimal places.

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 x > y +0.4.

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 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.

Find the mass of the solid cylinder $x ^ { 2 } + y ^ { 2 } \leq 25$ , $4 \leq Z \leq 6$ with density function $f ( x , y , z ) = z + x ^ { 2 } + y ^ { 2 }$

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

Upper and lower sums for a function f on a rectangle R, using n subdivisions on each side, are $\left( 3 n ^ { 2 } + 8 n + 24 \right) / n ^ { 2 }$ and $\left( 3 n ^ { 2 } + 7 n + 8 \right) / n ^ { 2 }$ respectively.Evaluate $\int _ { R } f d A$

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.

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

The function $f ( x , y ) = c x ^ { 2 } + 5 y$ has an average value of 4 on the triangle with vertices at (0, 0), (0, 1)and (1, 0).Find the constant a.

The function $f ( x ) = k x ^ { 2 } + 4 y$ 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.

Evaluate the integral $\int _ { 4 } ^ { 6 } \int _ { - 1 } ^ { - 1 } \int _ { - \sqrt { 16 - y ^ { 2 } } } ^ { \sqrt { 16 - y ^ { 2 } } } \frac { 1 } { \sqrt { x ^ { 2 } + y ^ { 2 } } } d x d y d z$ in cylindrical coordinates.

A Library of Functions

Key Concept: the Derivative

Short-Cuts to Differentiation

Using the Derivative

Key Concept- the Definite Integral

Constructing Antiderivatives

Integration

Using the Definite Integral

Sequences and Series

Approximating Functions Using Series

Differential Equations

Functions of Several Variables

A Fundamental Tool- Vectors

Differentiating Functions of Several Variables

Optimization- Local and Global Extrema

Parameterization and Vector Fields

Line Integrals

Flux Integrals and Divergence

The Curl and Stokes Theorem

Parameters, Coordinates, Integrals

Filters

Exam 16: Integrating Functions of Several Variables

Evaluate the integral ∫03∫−y/30ey2dxdy\int _ { 0 } ^ { 3 } \int _ { - y / 3 } ^ { 0 } e ^ { y ^ { 2 } } d x d y∫03​∫−y/30​ey2dxdy .Give your answer to two decimal places.

Set up (but do not evaluate)an iterated integral to compute the mass of the solid paraboloid bounded by z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2 and z = 1, if the density is given by δ\deltaδ (x, y, z)= z2.

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 δ\deltaδ (x, y, z)= z, find its mass.

Set up the three-dimensional integral ∫RydV\int _ { R } y d V∫R​ydV where R is the "ice-cream cone" enclosed by a sphere of radius 2 centered at the origin and the cone z=3x2+3y2z = \sqrt { 3 x ^ { 2 } + 3 y ^ { 2 } }z=3x2+3y2​ .Use rectangular coordinates.

Find the mass of the solid cylinder x2+y2≤25x ^ { 2 } + y ^ { 2 } \leq 25x2+y2≤25 , 4≤Z≤64 \leq Z \leq 64≤Z≤6 with density function f(x,y,z)=z+x2+y2f ( x , y , z ) = z + x ^ { 2 } + y ^ { 2 }f(x,y,z)=z+x2+y2

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

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.

Convert the integral to polar coordinates. ∫−11∫12−x2f(x,y)dydx\int _ { - 1 } ^ { 1 } \int _ { 1 } ^ { \sqrt { 2 - x ^ { 2 } } } f ( x , y ) d y d x∫−11​∫12−x2​​f(x,y)dydx

The function f(x,y)=cx2+5yf ( x , y ) = c x ^ { 2 } + 5 yf(x,y)=cx2+5y has an average value of 4 on the triangle with vertices at (0, 0), (0, 1)and (1, 0).Find the constant a.

The function f(x)=kx2+4yf ( x ) = k x ^ { 2 } + 4 yf(x)=kx2+4y 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.

A Library of Functions

Key Concept: the Derivative

Short-Cuts to Differentiation

Using the Derivative

Key Concept- the Definite Integral

Constructing Antiderivatives

Integration

Using the Definite Integral

Sequences and Series

Approximating Functions Using Series

Differential Equations

Functions of Several Variables

A Fundamental Tool- Vectors

Differentiating Functions of Several Variables

Optimization- Local and Global Extrema

Parameterization and Vector Fields

Line Integrals

Flux Integrals and Divergence

The Curl and Stokes Theorem

Parameters, Coordinates, Integrals

Filters

Evaluate the integral $\int _ { 0 } ^ { 3 } \int _ { - y / 3 } ^ { 0 } e ^ { y ^ { 2 } } d x d y$ .Give your answer to two decimal places.

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 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.

Find the mass of the solid cylinder $x ^ { 2 } + y ^ { 2 } \leq 25$ , $4 \leq Z \leq 6$ with density function $f ( x , y , z ) = z + x ^ { 2 } + y ^ { 2 }$

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

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

The function $f ( x , y ) = c x ^ { 2 } + 5 y$ has an average value of 4 on the triangle with vertices at (0, 0), (0, 1)and (1, 0).Find the constant a.

The function $f ( x ) = k x ^ { 2 } + 4 y$ 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.