Question 1

Find the exact area of the surface. $z = x ^ { 2 } + 2 y , 0 \leq x \leq 1,0 \leq y \leq 2$ .

Accepted Answer

A)  $\frac { 5 \ln ( 3 ) } { 4 }$ 
B)  $\frac { 5 } { 4 }$ 
C)  $3 - \ln ( 5 )$ 
D)  $3 + \frac { 5 } { 4 } \ln ( 5 )$ 
E)  $2 + \frac { 5 } { 3 } \ln ( 4 )$ 
A)  $\frac { 5 \ln ( 3 ) } { 4 }$ 
B)  $\frac { 5 } { 4 }$ 
C)  $3 - \ln ( 5 )$ 
D)  $3 + \frac { 5 } { 4 } \ln ( 5 )$ 
E)  $2 + \frac { 5 } { 3 } \ln ( 4 )$

Question 2

Find the average value of $f ( x , y ) = 2 x ^ { 2 } y$ over the rectangle with vertices $( - 1,0 ) , ( - 1,5 ) , ( 1,5 ) , ( 1,0 )$ .

Accepted Answer

The answer of Find the average value of \(f (...

Question 3

Use polar coordinates to find the volume of the sphere of radius $3$ . Round to two decimal places.

Accepted Answer

A)  $183.33$ 
B)  $113.1$ 
C)  $173.33$ 
D)  $153.33$ 
E)  $133.1$ 
A)  $183.33$ 
B)  $113.1$ 
C)  $173.33$ 
D)  $153.33$ 
E)  $133.1$

Question 4

Evaluate the double integral. $\iint _ { D } ( 4 x - y ) d A$ $D$ is bounded by the circle with center the origin and radius $36$ .

Accepted Answer

The answer of Evaluate the double integral. \(\iint _ {...

Question 5

Evaluate the double integral $$\iint _ { R } 3 x ^ { 2 } d A$$ , where $R$ is the region bounded by the graphs of $y = ( x - 1 ) ^ { 2 }$ and $y = - x + 3$ .

Accepted Answer

The answer of Evaluate the double integral \[\iint _ {...

Question 6

Sketch the solid bounded by the graphs of the equations $z = x ^ { 2 } + y ^ { 2 }$ and $z = 32 - x ^ { 2 } - y ^ { 2 }$ , and then use a triple integral to find the volume of the solid.

Accepted Answer

The answer of Sketch the solid bounded by the graphs...

Question 7

Find the mass of the lamina that occupies the region D and has the given density function, if D is bounded by the parabola $x = y ^ { 2 }$ and the line $y = x - 2$ . $\rho ( x , y ) = 3$

Accepted Answer

A)  $\frac { 3 } { 2 }$ 
B)  $2$ 
C)  27 
D)  $\frac { 27 } { 2 }$ 
E)  None of these 
A)  $\frac { 3 } { 2 }$ 
B)  $2$ 
C)  27 
D)  $\frac { 27 } { 2 }$ 
E)  None of these

Question 8

Set up, but do not evaluate, the iterated integral giving the mass of the solid T bounded by the cylinder $x ^ { 2 } + z ^ { 2 } = 9$ in the first octant and the plane $z + y = 3$ having mass density given by $\rho ( x , y , z ) = x y + z ^ { 4 }$

Accepted Answer

The answer of Set up, but do not evaluate, the...

Question 9

Evaluate the double integral. $\iint _ { D } x \cos y d A$ $D$ is bounded by $y = 0 , y = x ^ { 2 }$ and $x = 6$ .

Accepted Answer

The answer of Evaluate the double integral. \(\iint _ {...

Question 10

Calculate the iterated integral. $\int _ { 1 } ^ { 4 } \int _ { 0 } ^ { 3 } ( 1 + 4 x y ) d x d y$

Accepted Answer

The answer of Calculate the iterated integral. \(\int _ {...

Question 11

Find the area of the surface. The part of the surface $z = x y$ that lies within the cylinder $x ^ { 2 } - y ^ { 2 } = 4$ .

Accepted Answer

The answer of Find the area of the surface. The...

Question 12

Evaluate the iterated integral by converting to polar coordinates. Round the answer to two decimal places. $\int _ { - 3 } ^ { 3 } \int _ { 0 } ^ { \sqrt { 9 - y ^ { 2 } } } \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 / 2 } d x d y$ .

Accepted Answer

A)  $152.68$ 
B)  $5.65$ 
C)  $14.14$ 
D)  $48.6$ 
E)  $381.7$ 
A)  $152.68$ 
B)  $5.65$ 
C)  $14.14$ 
D)  $48.6$ 
E)  $381.7$

Question 13

Find the area of the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 25 z$ that lies inside the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ .

Accepted Answer

A)  $11.5 \pi$ 
B)  $9.5 \pi$ 
C)  $25 \pi$ 
D)  $5 \pi$ 
E)  $15 \pi$ 
A)  $11.5 \pi$ 
B)  $9.5 \pi$ 
C)  $25 \pi$ 
D)  $5 \pi$ 
E)  $15 \pi$

Question 14

Evaluate $\iint _ { D } 2 x ^ { 2 } y ^ { 2 } d A$ where $D$ is the figure bounded by $y = 1 , y = 2 , x = 0$ and $x = y$ .

Accepted Answer

The answer of Evaluate \(\iint _ { D } 2...

Question 15

Evaluate the iterated integral $\int _ { 0 } ^ { 4 } \int _ { 0 } ^ { 5 } \int _ { 0 } ^ { x } 3 \sqrt { y } e ^ { - x ^ { 2 } } d z d x d y$

Accepted Answer

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

Question 16

Evaluate the integral $\iiint _ { B } f ( x , y , z ) d V$ where $f ( x , y , z ) = x y ^ { 2 } + y z ^ { 2 }$ and $B = \{ ( x , y , z ) \mid 0 \leq x \leq 2 , - 5 \leq y \leq 5,0 \leq z \leq 3 \}$ with respect to x, y, and z, in that order.

Accepted Answer

A)  120 
B)  620 
C)  180 
D)  500 
A)  120 
B)  620 
C)  180 
D)  500

Question 17

Calculate the iterated integral. Round your answer to two decimal places. $\int _ { 0 } ^ { 6 } \int _ { 0 } ^ { 3 } \sqrt { 16 x + 16 y } d x d y$

Accepted Answer

A)  $188.52$ 
B)  $208.52$ 
C)  $168.52$ 
D)  $148.52$ 
E)  $158.52$ 
A)  $188.52$ 
B)  $208.52$ 
C)  $168.52$ 
D)  $148.52$ 
E)  $158.52$

Question 18

Find the Jacobian of the transformation. $$x = 6 \alpha \sin \beta , y = 5 \alpha \cos \beta$$

Accepted Answer

A)  $\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = - 30 \alpha$ 
B)  $\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = - 20 \alpha \sin \beta \cos \beta$ 
C)  $\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = 9 \alpha$ 
D)  $\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = - \alpha$ 
E)  $\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = 36 \alpha$ 
A)  $\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = - 30 \alpha$ 
B)  $\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = - 20 \alpha \sin \beta \cos \beta$ 
C)  $\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = 9 \alpha$ 
D)  $\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = - \alpha$ 
E)  $\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = 36 \alpha$

Question 19

Evaluate the integral by reversing the order of integration. $\int _ { 0 } ^ { 1 } \int _ { 6 y } ^ { 6 } e ^ { x ^ { 2 } } d x d y$

Accepted Answer

The answer of Evaluate the integral by reversing the order...

Question 20

Use cylindrical coordinates to evaluate $\int _ { - 2 } ^ { 2 } \int _ { 0 } ^ { \sqrt { 4 - x ^ { 2 } } } \int _ { 0 } ^ { \sqrt { 16 - x ^ { 2 } - y ^ { 2 } } } z d z d y d x$

Accepted Answer

A)  $12$ $\pi$ 
B)  $112$ $\pi$ 
C)  $\frac { 3 } { 2 }$ $\pi$ 
D)  $14$ $\pi$ 
A)  $12$ $\pi$ 
B)  $112$ $\pi$ 
C)  $\frac { 3 } { 2 }$ $\pi$ 
D)  $14$ $\pi$

Find the exact area of the surface. $z = x ^ { 2 } + 2 y , 0 \leq x \leq 1,0 \leq y \leq 2$ .

Find the average value of $f ( x , y ) = 2 x ^ { 2 } y$ over the rectangle with vertices $( - 1,0 ) , ( - 1,5 ) , ( 1,5 ) , ( 1,0 )$ .

Use polar coordinates to find the volume of the sphere of radius $3$ . Round to two decimal places.

Evaluate the double integral. $\iint _ { D } ( 4 x - y ) d A$ $D$ is bounded by the circle with center the origin and radius $36$ .

Evaluate the double integral $\iint _ { R } 3 x ^ { 2 } d A$ , where $R$ is the region bounded by the graphs of $y = ( x - 1 ) ^ { 2 }$ and $y = - x + 3$ .

Sketch the solid bounded by the graphs of the equations $z = x ^ { 2 } + y ^ { 2 }$ and $z = 32 - x ^ { 2 } - y ^ { 2 }$ , and then use a triple integral to find the volume of the solid.

Find the mass of the lamina that occupies the region D and has the given density function, if D is bounded by the parabola $x = y ^ { 2 }$ and the line $y = x - 2$ . $\rho ( x , y ) = 3$

Set up, but do not evaluate, the iterated integral giving the mass of the solid T bounded by the cylinder $x ^ { 2 } + z ^ { 2 } = 9$ in the first octant and the plane $z + y = 3$ having mass density given by $\rho ( x , y , z ) = x y + z ^ { 4 }$

Evaluate the double integral. $\iint _ { D } x \cos y d A$ $D$ is bounded by $y = 0 , y = x ^ { 2 }$ and $x = 6$ .

Calculate the iterated integral. $\int _ { 1 } ^ { 4 } \int _ { 0 } ^ { 3 } ( 1 + 4 x y ) d x d y$

Find the area of the surface. The part of the surface $z = x y$ that lies within the cylinder $x ^ { 2 } - y ^ { 2 } = 4$ .

Evaluate the iterated integral by converting to polar coordinates. Round the answer to two decimal places. $\int _ { - 3 } ^ { 3 } \int _ { 0 } ^ { \sqrt { 9 - y ^ { 2 } } } \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 / 2 } d x d y$ .

Find the area of the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 25 z$ that lies inside the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ .

Evaluate $\iint _ { D } 2 x ^ { 2 } y ^ { 2 } d A$ where $D$ is the figure bounded by $y = 1 , y = 2 , x = 0$ and $x = y$ .

Evaluate the iterated integral $\int _ { 0 } ^ { 4 } \int _ { 0 } ^ { 5 } \int _ { 0 } ^ { x } 3 \sqrt { y } e ^ { - x ^ { 2 } } d z d x d y$

Evaluate the integral $\iiint _ { B } f ( x , y , z ) d V$ where $f ( x , y , z ) = x y ^ { 2 } + y z ^ { 2 }$ and $B = \{ ( x , y , z ) \mid 0 \leq x \leq 2 , - 5 \leq y \leq 5,0 \leq z \leq 3 \}$ with respect to x, y, and z, in that order.

Calculate the iterated integral. Round your answer to two decimal places. $\int _ { 0 } ^ { 6 } \int _ { 0 } ^ { 3 } \sqrt { 16 x + 16 y } d x d y$

Find the Jacobian of the transformation. $x = 6 \alpha \sin \beta , y = 5 \alpha \cos \beta$

Evaluate the integral by reversing the order of integration. $\int _ { 0 } ^ { 1 } \int _ { 6 y } ^ { 6 } e ^ { x ^ { 2 } } d x d y$

Use cylindrical coordinates to evaluate $\int _ { - 2 } ^ { 2 } \int _ { 0 } ^ { \sqrt { 4 - x ^ { 2 } } } \int _ { 0 } ^ { \sqrt { 16 - x ^ { 2 } - y ^ { 2 } } } z d z d y d x$

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Exam 15: Multiple Integrals

Find the exact area of the surface. z=x2+2y,0≤x≤1,0≤y≤2z = x ^ { 2 } + 2 y , 0 \leq x \leq 1,0 \leq y \leq 2z=x2+2y,0≤x≤1,0≤y≤2 .

Find the average value of f(x,y)=2x2yf ( x , y ) = 2 x ^ { 2 } yf(x,y)=2x2y over the rectangle with vertices (−1,0),(−1,5),(1,5),(1,0)( - 1,0 ) , ( - 1,5 ) , ( 1,5 ) , ( 1,0 )(−1,0),(−1,5),(1,5),(1,0) .

Use polar coordinates to find the volume of the sphere of radius 333 . Round to two decimal places.

Evaluate the double integral. ∬D(4x−y)dA\iint _ { D } ( 4 x - y ) d A∬D​(4x−y)dA DDD is bounded by the circle with center the origin and radius 363636 .

Evaluate the double integral ∬R3x2dA\iint _ { R } 3 x ^ { 2 } d A∬R​3x2dA , where RRR is the region bounded by the graphs of y=(x−1)2y = ( x - 1 ) ^ { 2 }y=(x−1)2 and y=−x+3y = - x + 3y=−x+3 .

Sketch the solid bounded by the graphs of the equations z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2 and z=32−x2−y2z = 32 - x ^ { 2 } - y ^ { 2 }z=32−x2−y2 , and then use a triple integral to find the volume of the solid.

Find the mass of the lamina that occupies the region D and has the given density function, if D is bounded by the parabola x=y2x = y ^ { 2 }x=y2 and the line y=x−2y = x - 2y=x−2 . ρ(x,y)=3\rho ( x , y ) = 3ρ(x,y)=3

Set up, but do not evaluate, the iterated integral giving the mass of the solid T bounded by the cylinder x2+z2=9x ^ { 2 } + z ^ { 2 } = 9x2+z2=9 in the first octant and the plane z+y=3z + y = 3z+y=3 having mass density given by ρ(x,y,z)=xy+z4\rho ( x , y , z ) = x y + z ^ { 4 }ρ(x,y,z)=xy+z4

Evaluate the double integral. ∬Dxcos⁡ydA\iint _ { D } x \cos y d A∬D​xcosydA DDD is bounded by y=0,y=x2y = 0 , y = x ^ { 2 }y=0,y=x2 and x=6x = 6x=6 .

Calculate the iterated integral. ∫14∫03(1+4xy)dxdy\int _ { 1 } ^ { 4 } \int _ { 0 } ^ { 3 } ( 1 + 4 x y ) d x d y∫14​∫03​(1+4xy)dxdy

Find the area of the surface. The part of the surface z=xyz = x yz=xy that lies within the cylinder x2−y2=4x ^ { 2 } - y ^ { 2 } = 4x2−y2=4 .

Find the area of the part of the sphere x2+y2+z2=25zx ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 25 zx2+y2+z2=25z that lies inside the paraboloid z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2 .

Evaluate ∬D2x2y2dA\iint _ { D } 2 x ^ { 2 } y ^ { 2 } d A∬D​2x2y2dA where DDD is the figure bounded by y=1,y=2,x=0y = 1 , y = 2 , x = 0y=1,y=2,x=0 and x=yx = yx=y .

Evaluate the iterated integral ∫04∫05∫0x3ye−x2dzdxdy\int _ { 0 } ^ { 4 } \int _ { 0 } ^ { 5 } \int _ { 0 } ^ { x } 3 \sqrt { y } e ^ { - x ^ { 2 } } d z d x d y∫04​∫05​∫0x​3y​e−x2dzdxdy

Calculate the iterated integral. Round your answer to two decimal places. ∫06∫0316x+16ydxdy\int _ { 0 } ^ { 6 } \int _ { 0 } ^ { 3 } \sqrt { 16 x + 16 y } d x d y∫06​∫03​16x+16y​dxdy

Find the Jacobian of the transformation. x=6αsin⁡β,y=5αcos⁡βx = 6 \alpha \sin \beta , y = 5 \alpha \cos \betax=6αsinβ,y=5αcosβ

Evaluate the integral by reversing the order of integration. ∫01∫6y6ex2dxdy\int _ { 0 } ^ { 1 } \int _ { 6 y } ^ { 6 } e ^ { x ^ { 2 } } d x d y∫01​∫6y6​ex2dxdy

Use cylindrical coordinates to evaluate ∫−22∫04−x2∫016−x2−y2zdzdydx\int _ { - 2 } ^ { 2 } \int _ { 0 } ^ { \sqrt { 4 - x ^ { 2 } } } \int _ { 0 } ^ { \sqrt { 16 - x ^ { 2 } - y ^ { 2 } } } z d z d y d x∫−22​∫04−x2​​∫016−x2−y2​​zdzdydx

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Find the exact area of the surface. $z = x ^ { 2 } + 2 y , 0 \leq x \leq 1,0 \leq y \leq 2$ .

Find the average value of $f ( x , y ) = 2 x ^ { 2 } y$ over the rectangle with vertices $( - 1,0 ) , ( - 1,5 ) , ( 1,5 ) , ( 1,0 )$ .

Use polar coordinates to find the volume of the sphere of radius $3$ . Round to two decimal places.

Evaluate the double integral. $\iint _ { D } ( 4 x - y ) d A$ $D$ is bounded by the circle with center the origin and radius $36$ .

Evaluate the double integral $\iint _ { R } 3 x ^ { 2 } d A$ , where $R$ is the region bounded by the graphs of $y = ( x - 1 ) ^ { 2 }$ and $y = - x + 3$ .

Sketch the solid bounded by the graphs of the equations $z = x ^ { 2 } + y ^ { 2 }$ and $z = 32 - x ^ { 2 } - y ^ { 2 }$ , and then use a triple integral to find the volume of the solid.

Find the mass of the lamina that occupies the region D and has the given density function, if D is bounded by the parabola $x = y ^ { 2 }$ and the line $y = x - 2$ . $\rho ( x , y ) = 3$

Set up, but do not evaluate, the iterated integral giving the mass of the solid T bounded by the cylinder $x ^ { 2 } + z ^ { 2 } = 9$ in the first octant and the plane $z + y = 3$ having mass density given by $\rho ( x , y , z ) = x y + z ^ { 4 }$

Evaluate the double integral. $\iint _ { D } x \cos y d A$ $D$ is bounded by $y = 0 , y = x ^ { 2 }$ and $x = 6$ .

Calculate the iterated integral. $\int _ { 1 } ^ { 4 } \int _ { 0 } ^ { 3 } ( 1 + 4 x y ) d x d y$

Find the area of the surface. The part of the surface $z = x y$ that lies within the cylinder $x ^ { 2 } - y ^ { 2 } = 4$ .

Find the area of the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 25 z$ that lies inside the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ .

Evaluate $\iint _ { D } 2 x ^ { 2 } y ^ { 2 } d A$ where $D$ is the figure bounded by $y = 1 , y = 2 , x = 0$ and $x = y$ .

Evaluate the iterated integral $\int _ { 0 } ^ { 4 } \int _ { 0 } ^ { 5 } \int _ { 0 } ^ { x } 3 \sqrt { y } e ^ { - x ^ { 2 } } d z d x d y$

Calculate the iterated integral. Round your answer to two decimal places. $\int _ { 0 } ^ { 6 } \int _ { 0 } ^ { 3 } \sqrt { 16 x + 16 y } d x d y$

Find the Jacobian of the transformation. $x = 6 \alpha \sin \beta , y = 5 \alpha \cos \beta$

Evaluate the integral by reversing the order of integration. $\int _ { 0 } ^ { 1 } \int _ { 6 y } ^ { 6 } e ^ { x ^ { 2 } } d x d y$

Use cylindrical coordinates to evaluate $\int _ { - 2 } ^ { 2 } \int _ { 0 } ^ { \sqrt { 4 - x ^ { 2 } } } \int _ { 0 } ^ { \sqrt { 16 - x ^ { 2 } - y ^ { 2 } } } z d z d y d x$