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1.

$\int\int\int\left(2x-3y+4z\right)dxdydz$

a)

$xyz\left(x+\frac{3}{2}y+2z\right)+C$

b)

$xyz\left(x-3y+2z\right)+C$

c)

$xyz\left(x-\frac{3}{2}y+2z\right)+C$

d)

$xyz\left(x+3y+2z\right)+C$

2.

$\int\int\int\left(x^2+y^2+z^2\right)dxdydz$

a)

$3xyz\left(x^2yz+xy^2z+xyz^2\right)+C$

b)

$\frac{1}{3}xyz\left(x^2+y^2+z^2\right)+C$

c)

$xyz\left(x^2yz+xy^2z+xyz^2\right)+C$

d)

$\frac{1}{3}xyz\left(x^2y^2z+xy^2z^2+x^2yz^2\right)+C$

3.

$\int\int\int e^{4x}dxdydz$

a)

$\frac{1}{3}e^{4xy}yz+C$

b)

$\frac{1}{3}e^{4x}yz+C$

c)

$\frac{1}{4}e^{4x}xyz+C$

d)

$\frac{1}{4}e^{4x}yz+C$

4.

$\int_0^1\int_1^2\int_3^4\ \left(-x-y-z\right)dxdydz$

a)

5.50

b)

-5.50

c)

8.50

d)

-8.50

5.

$\int_2^4\int_6^8\int_{-2}^{-4}\ \cos\left(y\right)\ dxdydz$

a)

$-8.05$

b)

$\approx-0.192$

c)

$\approx-2.19$

d)

$\approx-5.08$

6.

$\int_2^4\int_6^8\int_{-2}^{-4}\ \left(e^z\right)\ dxdydz$

a)

$\approx-189$

b)

$\approx-132$

c)

$\approx538$

d)

$\approx710$

7.

Find the lower limit of z 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,

taking the order of integration as

dzdydx

.

a)

$5$

b)

$y-6$

c)

$4-y$

d)

$0$

8.

Find the upper limit of z 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,

taking the order of integration as

dzdydx

.

a)

$5$

b)

$y-6$

c)

$4-y$

d)

$0$

9.

Find the upper limit of y 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,

taking the order of integration as

dzdydx

.

a)

$x^2$

b)

$5$

c)

$z+6$

d)

$0$

10.

$\int\int\int\left(2x-3y+4z\right)dxdydz$

a)

$xyz\left(x+\frac{3}{2}y+2z\right)+C$

b)

$xyz\left(x-3y+2z\right)+C$

c)

$xyz\left(x-\frac{3}{2}y+2z\right)+C$

d)

$xyz\left(x+3y+2z\right)+C$

11.

$\int\int\int\left(x^2+y^2+z^2\right)dxdydz$

a)

$3xyz\left(x^2yz+xy^2z+xyz^2\right)+C$

b)

$\frac{1}{3}xyz\left(x^2+y^2+z^2\right)+C$

c)

$xyz\left(x^2yz+xy^2z+xyz^2\right)+C$

d)

$\frac{1}{3}xyz\left(x^2y^2z+xy^2z^2+x^2yz^2\right)+C$

12.

$\int\int\int e^{4x}dxdydz$

a)

$\frac{1}{3}e^{4xy}yz+C$

b)

$\frac{1}{3}e^{4x}yz+C$

c)

$\frac{1}{4}e^{4x}xyz+C$

d)

$\frac{1}{4}e^{4x}yz+C$

13.

$\int_0^1\int_1^2\int_3^4\ \left(-x-y-z\right)dxdydz$

a)

5.50

b)

-5.50

c)

8.50

d)

-8.50

14.

$\int_2^4\int_6^8\int_{-2}^{-4}\ \cos\left(y\right)\ dxdydz$

a)

$-8.05$

b)

$\approx-0.192$

c)

$\approx-2.19$

d)

$\approx-5.08$

15.

$\int_2^4\int_6^8\int_{-2}^{-4}\ \left(e^z\right)\ dxdydz$

a)

$\approx-189$

b)

$\approx-132$

c)

$\approx538$

d)

$\approx710$

16.

Find the lower limit of z 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,

taking the order of integration as

dzdydx

.

a)

$5$

b)

$y-6$

c)

$4-y$

d)

$0$

17.

Find the upper limit of z 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,

taking the order of integration as

dzdydx

.

a)

$5$

b)

$y-6$

c)

$4-y$

d)

$0$

18.

Find the lower limit of y 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,

taking the order of integration as

dzdydx

.

a)

$x^2$

b)

$5$

c)

$z+6$

d)

$0$

19.

Find the upper limit of y 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,

taking the order of integration as

dzdydx

.

a)

$x^2$

b)

$5$

c)

$z+6$

d)

$0$

20.

Find the lower limit of x 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,

taking the order of integration as

dzdydx

.

a)

$5$

b)

$-5$

c)

$\sqrt{5}$

d)

$-\sqrt{5}$

21.

Find the upper limit of x 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,

taking the order of integration as

dzdydx

.

a)

$5$

b)

$-5$

c)

$\sqrt{5}$

d)

$-\sqrt{5}$

22.

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)

23.

Determine the intervals for $r$ and $\theta$ associated with the blue region.

a)

$3\le r\le4,\ 0\le\theta\le\pi$

b)

$2\le r\le4,\ 0\le\theta\le\frac{\pi}{4}$

c)

$\pi\le r\le\frac{\pi}{4},\ 1\le\theta\le4$

d)

$2\le r\le4,\ \frac{\pi}{4}\le\theta\le\pi$

24.

Convert $\int_0^{\frac{1}{\sqrt{2}}}\int_x^{\sqrt{1-x^2}}\left(3x\ +4y^2\right)\ dydx$ into polar coordinates

a)

$\int_0^{\frac{\pi}{4}}\int_0^13r+4r\sin\theta\ drd\theta$

b)

$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\int_0^1r^2\left(3\cos\theta+4r\sin^2\theta\right)drd\theta$

c)

$\int_0^{\frac{\pi}{2}}\int_0^13r\cos\theta+4r^2\sin^2\theta\ drd\theta$

d)

$\int_{\frac{\pi}{3}}^{\frac{\pi}{2}}\int_0^13\cos\theta+4r^2\sin^2\theta\ r\ drd\theta$

25.

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)

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

26.

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

27.

Determine the intervals for $r$ and $\theta$ associated with the blue region.

a)

$3\le r\le4,\ 0\le\theta\le\pi$

b)

$2\le r\le4,\ 0\le\theta\le\frac{\pi}{4}$

c)

$\pi\le r\le\frac{\pi}{4},\ 1\le\theta\le4$

d)

$2\le r\le4,\ \frac{\pi}{4}\le\theta\le\pi$

28.

Find the upper limit of z 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,

taking the order of integration as

dzdydx

.

a)

$5$

b)

$y-6$

c)

$4-y$

d)

$0$

29.

Find the upper limit of z 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,

taking the order of integration as

dzdydx

.

a)

$5$

b)

$y-6$

c)

$4-y$

d)

$0$

30.

Find the lower limit of z 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,

taking the order of integration as

dzdydx

.

a)

$5$

b)

$y-6$

c)

$4-y$

d)

$0$

31.

Find the upper limit of y 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,

taking the order of integration as

dzdydx

.

a)

$x^2$

b)

$5$

c)

$z+6$

d)

$0$

32.

Find the lower limit of y 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,

taking the order of integration as

dzdydx

.

a)

$x^2$

b)

$5$

c)

$z+6$

d)

$0$

33.

Find the upper limit of x 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,

taking the order of integration as

dzdydx

.

a)

$5$

b)

$-5$

c)

$\sqrt{5}$

d)

$-\sqrt{5}$

34.

Find the lower limit of x 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,

taking the order of integration as

dzdydx

.

a)

$5$

b)

$-5$

c)

$\sqrt{5}$

d)

$-\sqrt{5}$

35.

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)

36.

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

a)

1

b)

0

c)

$\pi$

d)

$\frac{\pi}{2}$

37.

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

a)

$e^2-e$

b)

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

c)

$e^2$

d)

$e^2-1$

38.

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)

39.

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)

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

40.

$\int_0^2\int_0^{\infty}e^{-x}dxdy$ is ___________

a)

2

b)

1

c)

$e^2-1$

d)

$e^{-2}-1$

41.

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

a)

$e^2-e$

b)

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

c)

$e^2$

d)

$e^2-1$

42.

$\int_0^2\int_0^yx^2dxdy$ is ___________

a)

$\frac{2}{3}$

b)

$\frac{4}{3}$

c)

$\frac{1}{3}$

d)

$\frac{5}{3}$

43.

¿ Cuál de las siguientes imágenes es una coordenada cilíndrica?

a)

b)

c)

d)

44.

Región determinada por

$2\le r\le4\ ,\ 0\le z\le3\ ,\ \ 0\le\theta\le\frac{\pi}{2}\$

a)

b)

c)

d)

45.

$\int_0^1\int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}\ \int_{x^2+y^2}^4\ \ \ \ x\ dzdydx$ . Si a la integral dada le realizamos un cambio de variable a coordenadas cilíndricas obtenemos:

a)

$\int_0^{2\pi}\int_0^2\int_{r^2}^4\ \ r^2sen\theta\ dzdrd\theta$

b)

$\int_0^{2\pi}\int_0^2\int_{r^2}^4\ \ r^2\cos\theta\ dzdrd\theta$

c)

$\int_0^2\int_0^{2\pi}\int_{r^2}^4\ \ r^2\cos\theta\ dzdrd\theta$

d)

$\int_0^{2\pi}\int_0^2\int_{r^2}^4\ \ r^2\cos\theta\ dzd\theta dr$

46.

Describir la región dada en coordenadas esféricas

a)

$E=\left\{0\le\theta\le2\pi;\ 0\le\phi\le\frac{\pi}{2};\ 9\le\rho\le25\right\}$

b)

$E=\left\{0\le\theta\le2\pi;\ 0\le\phi\le\frac{\pi}{4};\ 3\le\rho\le5\right\}$

c)

$E=\left\{0\le\theta\le2\pi;\ 0\le\phi\le\frac{\pi}{3};\ 3\le\rho\le5\right\}$

47.

Se tiene el sólido E encerrada por las superficies $S_1:\ z=\sqrt{8-x^2-y^2\ }\ \ ;\ \ \ S_2:\ z\ =\sqrt{x^2+y^2}$ . La descripción del solido en coordenadas esféricas es::

a)

$E=\left\{0\le\theta\le2\pi;\ 0\le\phi\le\frac{\pi}{4};\ 0\le\rho\le\sqrt{8}\right\}$

b)

$E=\left\{0\le\theta\le2\pi;\ 0\le\phi\le\frac{\pi}{2};\ 0\le\rho\le8\right\}$

c)

$E=\left\{0\le\theta\le\frac{\pi}{2};\ 0\le\phi\le\frac{\pi}{2};\ 0\le\rho\le8\right\}$

48.

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)

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

49.

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)

50.

Evaluate the triple integral

$\int\int\int_E6xy\ dV$ where $E$ lies under the plane $z=1+x+y$ and above the region in the xy-plane bounded by the curves $y=\sqrt{x\ },\ y=0,\ x=1$ . The result is equal to (a) (Please write in a fraction term a/b)

51.

Calculate the value of the multiple integral $\int_{ }^{ }\int_D^{ }\left(x^2+y^2\right)^{\frac{3}{2}}dA$ , where $D$ is the region in the first quadrant bounded by the lines $y=0$ and $y=\sqrt[]{3}x$ and the circle $x^2+y^2=9$ . The result is equal to___

a)

$\frac{81\pi}{5}$

b)

$\frac{27\pi}{5}$

c)

$\frac{9\pi}{5}$

d)

$\frac{3\pi}{5}$

52.

Is it TRUE or FALSE:

$\int_{-1}^2\int_0^5x^2\sin\left(x-y\right)dxdy=\int_0^5\int_{-1}^2x^2\sin\left(x-y\right)dydx$

a)

TRUE

b)

FALSE

53.

What is the formula for calculating the volume of a solid using a triple integral?

a)

∭∭∭ f(x, y, z) dV

b)

∬∬∬ f(x, y, z) dV

c)

∭∭ f(x, y, z) dV

d)

∫∫∫ f(x, y, z) dV

54.

What is the concept of iterated integrals?

a)

A mathematical concept used to calculate multiple integrals by breaking the region into smaller pieces and integrating over each piece.

b)

A cooking technique used to prepare multiple dishes simultaneously

c)

A concept in computer programming for repeating a sequence of instructions

d)

A method for solving algebraic equations using repeated addition

55.

How do you determine the limits of integration for a double integral?

a)

Guess randomly

b)

Consider the region of integration and the bounds of the variables involved.

c)

Use the value of the function at a single point

d)

Ask someone else to determine the limits

56.

Calcula el área bajo la curva

y=8x\left(x^2-1\right)^{10}

comprendida entre x=0 y x=1.

a)

$\frac{4}{11}u^2$

b)

$-\frac{2}{5}u^2$

c)

$-\frac{8}{11}u^2$

d)

$\frac{2}{5}u^2$

57.

Determine el valor de la integral que se muestra en la imagen

a)

200

b)

49

c)

15

d)

175

58.

Responder: ¿Qué es un integral definida?

a)

Una función

b)

Una antiderivada

c)

Un diferencial

d)

Un límite de sumas

59.

El valor de la integral $\int_{-1}^1x^3dx$ es:

a)

Positivo

b)

Negativo

c)

Cero

d)

No se puede determinar

60.

a)

$10.5$

b)

$21$

c)

$11$

d)

$12.5$

61.

Una suma de Riemann que permite calcular la integral $\int_1^2\left(3x^2-2x\right)dx$ es:

a)

$\lim_{n\rightarrow\infty}\sum_{k=1}^{\infty}\left(3\ \frac{k^2}{n^2}-2\ \frac{k}{n}\right)\ \frac{1}{n}$

b)

$\lim_{n\rightarrow\infty}\sum_{k=1}^{\infty}\left(\frac{1}{n}+4\ \frac{k}{n^2}+3\ \frac{k^{2\ }}{n^3}\right)$

c)

$\lim_{n\rightarrow\infty}\sum_{k=1}^{\infty}\left(3\ \left(1+\frac{k}{n}\right)^2+2\ \left(1+\frac{k}{n}\right)\right)\ \frac{1}{n}$

d)

Ninguna de las anteriores

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Total questions: 61

Worksheet time: 3570secs

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