Free Printable multiple integration

1.

$\int_0^{\frac{\pi}{2}}\int_0^1rdrd\theta$ is ___________

a)

0

b)

1

c)

$\frac{\pi}{4}$

d)

$\frac{\pi}{2}$

2.

$\int_0^3\int_0^3\int_0^3dxdydz$ is________

a)

3

b)

6

c)

9

d)

27

3.

$\int_0^1\int_0^xf\left(x,y\right)dydx=_{ }$ _________ ( BY change the order of integration)

a)

$\int_y^1\int_0^1xydxdy$

b)

$\int_0^x\int_0^1f\left(x,y\right)dydx$

c)

$\int_0^1\int_y^1f\left(x,y\right)dxdy$

d)

$\int_1^0\int_x^0f\left(x,y\right)dydx$

4.

The equal double integral in polar co-ordinates of $\int_0^{\infty}\int_0^{\infty}e^{-(x^2+y^2)}dxdy$ is ___________

a)

$\int_0^{\pi/2}\int_0^{\infty}e^{-r^2}rdrd\theta$

b)

$\int_0^{\pi/2}\int_0^1e^{-r^2}drd\theta$

c)

$\int_0^{\infty}\int_0^{\infty}e^{-r^2}drd\theta$

d)

$\int_0^{\infty}\int_0^{\infty}e^{-(r^2+\theta^2)}drd\theta$

5.

$\int_0^{\pi/2}\int_0^{\pi/2}\cos\theta\cos\varphi d\theta d\varphi$ is ___________

a)

1

b)

0

c)

$\pi$

d)

$\frac{\pi}{2}$

6.

In $\int_0^1\int_0^{1-z}\int_0^{1-y-z}xyzdxdydz$ the limit of x varies from____________

a)

0 to 1

b)

0 to 1-z

c)

0 to 1-y-z

d)

1 to 1-y-z

7.

$\int_1^2\int_0^2e^xdydx$ is ___________

a)

$e^2-e$

b)

$2\left(e^2-e\right)$

c)

$e^2$

d)

$e^2-1$

8.

$\int_2^1\int_0^1\left(x+y\right)^2dxdy$ =

a)

$-\frac{17}{3}$

b)

-6

c)

$-\frac{25}{6}$

d)

$\frac{39}{6}$

9.

Curves $x^2=2-y$ and $x=y$ are shown in this figure. Set up a double integral to find the area of the blue region.

a)

$\int_{-2}^1\int_x^{x^2}1\ dydx$

b)

$\int_x^2\int_1^21\ dydx$

c)

$\int_{-2}^1\int_x^{\left(2-x^2\right)}1\ dydx$

d)

$\int_{-2}^0\int_y^{\left(2-y\right)}1\ dxdy$

10.

Find the volume of the solid D (leave your answer correct to two decimal places) in the triple integral

\int_{ }^{ }\int_{ }^{ }\int_D^{ }\ \ f\left(x,y,z\right)dV

if D is the solid bounded by

z=4-y,\ z=y-6

and

y=x^2.

(a)

11.

Find the cylindrical coordinate for the Cartesian coordinate

\left(x,y,z\right)=\left(1,-1,\ 3\right).

a)

$\left(\sqrt{2},\frac{\pi}{4},3\right)$

b)

$\left(\sqrt{2},-\frac{\pi}{4},3\right)$

c)

$\left(\sqrt{11},\frac{\pi}{4},3\right)$

d)

$\left(\sqrt{11},-\frac{\pi}{4},3\right)$

12.

Set up the limit of triple integral to find the volume of the solid bounded by $z=4-x^2$ , $y+z=5\ ,z=0,\ y=0$

a)

b)

c)

d)

13.

Set up limit of triple integral to find the volume of the solid that lies below the hemisphere $z=\sqrt{25-x^2-y^2}$ between cylinder $x^2+y^2=16$ and $x^2+y^2=25$ using cylindrical coordinates.

a)

$\int_0^{2\pi}\int_0^4\int_0^{\sqrt{25-r^2}}rdzdrd\theta$

b)

$\int_0^{2\pi}\int_0^5\int_0^{\sqrt{25-r^2}}rdzdrd\theta$

c)

$\int_0^{2\pi}\int_4^5\int_0^{\sqrt{25-r^2}}rdzdrd\theta$

d)

$\int_0^{2\pi}\int_4^5\int_0^5rdzdrd\theta$

14.

What is dV in the triple integral in spherical coordinate system?

a)

$d\rho d\phi d\theta$

b)

$\rho^2\cos\phi\ d\rho d\phi d\theta$

c)

$\rho^2\sin\phi\ d\rho d\phi d\theta$

15.

Find the volume of the solid bounded by $z=x^2+y^2-2,\ z=10-x^2-y^2$ and $x^2+y^2=1$ . Leave your answer correct to two decimal places.

(a)

multiple integration

Total questions: 15

Worksheet time: 8mins

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