Question 1

Find an integral equal to the volume of the solid bounded by the given surfaces and evaluate the integral.

Accepted Answer

The answer of Find an integral equal to the volume...

Question 2

Evaluate the iterated integral by first changing the order of integration.

Accepted Answer

C

Question 3

Find the volume of the solid bounded by the given surfaces.

Accepted Answer

D

Question 4

Approximate the double integral.  , where R is bounded by x = 0, x = 3, y = 0, and y = x + 2

Accepted Answer

We can set up the integral as follows:

$\int_0^3 \int_0^{x+2} 11e^{x+y} dy dx$

Then we can evaluate the integral:

$\int_0^3 \int_0^{x+2} 11e^{x+y} dy dx = 11\int_0^3 e^x \int_0^{x+2} e^y dy dx = 11\int_0^3 e^x (e^{x+2} - e^x) dx$

$= 11\int_0^3 e^{3x+2} - e^{2x} dx = \frac{11}{3}(e^8 - e^2) \approx 2.658$

Therefore, the best choice is B.

Question 5

Compute the volume of the solid bounded by the given surfaces.  and the three coordinate planes

Accepted Answer

The answer of Compute the volume of the solid bounded...

Question 6

Evaluate the double integral.

Accepted Answer

The answer of Evaluate the double integral. 
...

Question 7

Evaluate the iterated integral by first changing the order of integration.

Accepted Answer

The answer of Evaluate the iterated integral by first changing...

Question 8

Use a double integral to find the area of the region bounded by  ,  and  .

Accepted Answer

The answer of Use a double integral to find the...

Question 9

Evaluate the iterated integral by first changing the order of integration.

Accepted Answer

The answer of Evaluate the iterated integral by first changing...

Question 10

Evaluate the double integral.

Accepted Answer

The answer of Evaluate the double integral. 
...

Question 11

Compute the Riemann sum for the given function and region, a partition with n equal-sized rectangles and the given evaluation rule.

Accepted Answer

The answer of Compute the Riemann sum for the given...

Question 12

Compute the Riemann sum for the given function, the irregular partition shown and midpoint evaluation.

Accepted Answer

The answer of Compute the Riemann sum for the given...

Question 13

Change the order of integration.

Accepted Answer

The answer of Change the order of integration. 
...

Question 14

Change the order of integration.

Accepted Answer

The answer of Change the order of integration. 
...

Question 15

Evaluate the iterated integral by first changing the order of integration.

Accepted Answer

The integral is given by 
$$\int_0^2 \int_{x/2}^x e^{y^2} \ dy \ dx.$$
We sketch the region of integration in the $xy$-plane (shaded in orange), which is a triangle bounded by $y=x/2$, $y=x$, and $x=2$.

Now, we change the order of integration. The region of integration is bounded by $0 \leq x \leq 2$ and $y/2 \leq x \leq 2$. Therefore, we have
\begin{align*}
\int_0^2 \int_{x/2}^x e^{y^2} \ dy \ dx &= \int_0^1 \int_0^{2y} e^{y^2} \ dx \ dy + \int_1^2 \int_{y/2}^y e^{y^2} \ dx \ dy \
&= \int_0^1 2ye^{y^2} \ dy + \int_1^2 (y- y/2) e^{y^2} \ dy \
&= e-1 + \frac{1}{2}(e^4 - e) \
&= \boxed{	extbf{(D) } 11eaa948_cd7f_1ab7_84bc_0df0fceaa776_TB2342_11}.
\end{align*}

Question 16

Evaluate the iterated integral.

Accepted Answer

The answer of Evaluate the iterated integral. 
...

Question 17

Find the volume of the solid bounded by the given surfaces.

Accepted Answer

The answer of Find the volume of the solid bounded...

Question 18

Evaluate the iterated integral.

Accepted Answer

The answer of Evaluate the iterated integral. 
...

Question 19

Evaluate the iterated integral.

Accepted Answer

The answer of Evaluate the iterated integral. 
...

Question 20

Use a double integral to find the area of the region bounded by  and  .

Accepted Answer

The answer of Use a double integral to find the...

Quiz 14: Multiple Integrals

Find an integral equal to the volume of the solid bounded by the given surfaces and evaluate the integral.

Evaluate the iterated integral by first changing the order of integration.

Find the volume of the solid bounded by the given surfaces.

Approximate the double integral. , where R is bounded by x = 0, x = 3, y = 0, and y = x + 2

Compute the volume of the solid bounded by the given surfaces. and the three coordinate planes

Evaluate the double integral.

Evaluate the iterated integral by first changing the order of integration.

Use a double integral to find the area of the region bounded by , and .

Evaluate the iterated integral by first changing the order of integration.

Evaluate the double integral.

Compute the Riemann sum for the given function and region, a partition with n equal-sized rectangles and the given evaluation rule.

Compute the Riemann sum for the given function, the irregular partition shown and midpoint evaluation.

Change the order of integration.

Change the order of integration.

Evaluate the iterated integral by first changing the order of integration.

Evaluate the iterated integral.

Find the volume of the solid bounded by the given surfaces.

Evaluate the iterated integral.

Evaluate the iterated integral.

Use a double integral to find the area of the region bounded by and .