Double and Triple Integrals

Evaluate Each of the integrals in exercises 33-36 as an iterated integral and then compare your answer with thoise you found in exercise 29-32

$$\begin{gathered} \int {\int }_R{x}^{2}ydA \\ Where R=\left\{\right(x,y\left)\right|-1\le x\le 0,0\le y\le 2\} \end{gathered}$$

Evaluate Each of the integrals in exercises 33-36 as an iterated integral and then compare your answer with thoise you found in exercise 29-32 

$$\begin{gathered} \int {\int }_{R}x{y}^{3}dA \\ Where R=\left\{\right(x,y\left)\right|-2\le x\le 2,-1\le y\le 1\} \end{gathered}$$

Evaluate each of the integrals in exercise 33-36 as iterated integrals and then compare your answers with those you found in exercise 29-32

$$\begin{gathered} \int {\int }_R{x}^2{y}^{3}dA \\ Where R=\left\{\right(x,y\left)\right|1\le x\le 3,0\le y\le 2\} \end{gathered}$$

Evaluate each of the double integral in the exercise 37-54 as iterated integrals

$$\begin{gathered} \int {\int }_R(3-x+4y)dA \\ where R=\left\{\right(x,y\left)\right|0\le x\le 1,-1\le y\le 3\} \end{gathered}$$

Evaluate each of the double integral in the exercise 37-54 as iterated integrals 

$$\begin{gathered} \int {\int }_R\left(y^2\right)dA \\ Where R=\left\{\right(x,y\left)\right|-3\le x\le 2,-2\le y\le 2\} \end{gathered}$$

Evaluate each of the double integral in the exercise 37-54 as iterated integrals  

$$\begin{gathered} \int {\int }_R(2-3x^2+y^3)dA \\ Where R=\left\{\right(x,y\left)\right|-3\le x\le 2,3\le y\le 5\} \end{gathered}$$

Evaluate each of the double integral in the exercise 37-54 as iterated integrals   

$$\begin{gathered} \int {\int }_R(x-e^y)dA \\ Where R=\left\{\right(x,y\left)\right|-3\le x\le 2,-2\le y\le 2\} \end{gathered}$$

Evaluate each of the double integral in the exercise 37-54 as iterated integrals

$$\begin{gathered} \int {\int }_R\sin (x+2y)dA \\ Where R=\left\{\right(x,y\left)\right|0\le x\le \pi ,0\le y\le \frac{\pi }{2}\} \end{gathered}$$

Evaluate each of the double integrals in Exercises 37–54 as iterated integrals.

$$\begin{gathered} \int {\int }_{R}x\sin x \cos y dA, \\ where R=\left\{\right(x,y\left)\right|-3\le x\le 2and-2\le y\le 2\} \end{gathered}$$

Evaluate each of the double integrals in Exercises 37–54 as iterated integrals.

$$\begin{gathered} \int {\int }_{R}x{e}^{xy} dA, \\ where \{R=(x,y)| 0\le x\le 1 and 0\le y\le ln5\} \end{gathered}$$

Evaluate each of the double integrals in Exercises 37–54 as iterated integrals. 

$$\begin{gathered} \int {\int }_R{x}^2\cos \left(xy\right) dA, \\ where R=\left\{\right(x,y\left)\right| 0\le x\le \pi and 0\le y\le 1\} \end{gathered}$$

Evaluate each of the double integrals in Exercises 37–54 as iterated integrals. 

$$\begin{gathered} \int {\int }_R\frac{x}{x+y}dA, \\ whereR=\left\{\right(x,y) | 1\le x\le 4 and 0\le y\le 3\} \end{gathered}$$

Evaluate each of the double integrals in Exercises 37–54 as iterated integrals. 

$$\begin{gathered} \int {\int }_R{x}^3{e}^{x^{2}y}dA, \\ where R=\left\{\right(x,y\left)\right| 0\le x\le 4 and -1\le y\le 1\} \end{gathered}$$

Evaluate each of the double integrals in Exercises $37-54$ as iterated integrals.

$\int {\int }_{R}y\sin x dA,$

where  $R=\left\{\left(x,y\right)|0\le x\le \frac{\mathrm{\pi }}{2} and 0\le y\le 1\right\}$

Evaluate each of the double integrals in Exercises $37-54$ as iterated integrals.

$\int {\int }_R\sin \left(2x+y\right)dA,$

where $R=\left\{\left(x,y\right)|0\le x\le \frac{\mathrm{\pi }}{2} and 0\le y\le \frac{\mathrm{\pi }}{2}\right\}$

Evaluate each of the double integrals in Exercises $37-54$ as iterated integrals.

$\int {\int }_R{y}^2\sin xdA,$

where $R=\left\{\left(x,y\right)|0\le x\le \mathrm{\pi } \mathrm{and} 0\le \mathrm{y}\le 3\right\}$

Evaluate each of the double integrals in Exercises $37-54$ as iterated integrals.

$\int {\int }_{R}xy\sin \left(x^2\right)dA,$

where $R=\left\{\left(x,y\right)|0\le x\le \sqrt{\mathrm{\pi }} and 0\le y\le 1\right\}$

Evaluate each of the double integrals in Exercises  $37-54$ as iterated integrals .

$\int {\int }_{R}y\cos \left(xy\right)dA,$

where $R=\left\{\left(x,y\right)|0\le x\le \frac{\mathrm{\pi }}{2} and 0\le y\le 1\right\}$

Evaluate each of the double integrals in Exercises $37-54$ as iterated integrals.

$\int {\int }_R{e}^{x+y}dA,$

where $R=\left\{\left(x,y\right)|0\le x\le 1 and 0\le y\le 1\right\}$.

Evaluate each of the double integrals in Exercises $37-54$as iterated integrals.

$\int {\int }_R{x}^2{e}^{xy}dA,$

where $R=\left\{\left(x,y\right)|0\le x\le 1 and 0\le y\le 1\right\}$.

Evaluate each of the double integrals in Exercises 37–54 as iterated integrals.

$\int {\int }_R\frac{dA}{x^2+2xy+y^2},$

where $R=\left\{\left(x,y\right)|1\le x\le 2 and 0\le y\le 1\right\}$.

In Exercises $55-58$, find the signed volume between the graph of the given function and the $xy$-plane over the specified rectangle in the$xy$-plane.

$f\left(x,y\right)=3x-2y^5+1$,

where $R=\left\{\left(x,y\right)|-4\le x\le 6 and 0\le y\le 7\right\}$

In Exercises $55-58$, find the signed volume between the graph of the given function and the $xy$-plane over the specified rectangle in the $xy$-plane.

$f\left(x,y\right)=x^2-y^3+4$,

where $R=\left\{\left(x,y\right)|0\le x\le 3 and -2\le y\le 3\right\}$.

Find the signed volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane 

$$\begin{gathered} f(x,y)=y^3{e}^{x{y}^2} \\ Where R=\left\{\right(x,y\left)\right|0\le x\le 2,-2\le y\le 3\} \end{gathered}$$

Find the signed volume between the graph of the function and the xy-plane over the specified region

$$\begin{gathered} f(x,y)=x{y}^3{e}^{x^2{y}^2} \\ Where R=\left\{\right(x,y\left)\right|-2\le x\le -1,-2\le y\le 0\} \end{gathered}$$

Find the signed volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane

$$\begin{gathered} f(x,y)=xy \\ Where R=\left\{\right(x,y\left)\right|-2\le x\le 3,-1\le y\le 5\} \end{gathered}$$

Find the volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane

$$\begin{gathered} f(x,y)=-2x^2{y}^3 \\ Where R=\left\{\right(x,y\left)\right|-2\le x\le 3,-1\le y\le 5\} \end{gathered}$$

Find the volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane 

$$\begin{gathered} f(x,y)=\sin x\cos y \\ Where R=\left\{\right(x,y\left)\right|0\le x\le \pi ,0\le y\le \pi \} \end{gathered}$$

Find the volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane  

$$\begin{gathered} f(x,y)=\frac{y}{x}{e}^y \\ Wjere R=\left\{\right(x,y\left)\right|1\le x\le e,0\le y\le 2\} \end{gathered}$$

Find the volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane 

$$\begin{gathered} f(x,y)=\frac{x}{y}+\frac{y}{x} \\ Where R=\left\{\right(x,y\left)\right|1\le x\le 3,1\le y\le 5\} \end{gathered}$$

Find the volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane  

$$\begin{gathered} f(x,y)=x^{2}y{e}^{xy} \\ Where R=\left\{\right(x,y\left)\right|0\le x\le 1,0\le y\le 2\} \end{gathered}$$

Use midpoint Riemann sums with the specified numbers

${\int }_0^1{\int }_0^{1}y\sin \left(x^2\right)dxdy.$  Let each sub rectangle be a square

with side length $\frac{1}{3}$unit.

Let $a<b\text{ and }c<d$ be real numbers and $R$ be rectangle defined by

$R=\left\{\right(x,y)\mid a\le x\le b\text{ and }c\le y\le d\}$ in $xy$ plane. If $g\left(x\right)$ is continuous in interval $[a, b]$ and $h\left(y\right)$ is continuous on $[c, d]$ 

Use Fubini's theorem to prove that ${\iint }_{R}g\left(x\right)h\left(y\right)dA=\left({\int }_a^{b}g\left(x\right)dx\right)\left({\int }_c^{d}h\left(y\right)dy\right).$

Let $a<b$ and $c<d$ be real numbers, and let $R$be the

rectangle in the $xy$-plane defined by

$R=\left\{\right(x,y)\mid a\le x\le b\text{ and }c\le y\le d\}.$

Prove that ${\iint }_{R}dA=(b-a)(d-c)$, what is the relation between $R$ and product of $(b-a)(d-c)$?

Explain the difference between a type I region and a type II region.

Let $R=\left\{\right(x,y)\mid a\le x\le b\text{ and }c\le y\le d\}$be a rectangular region. Explain why R is both a type I region and a type II region.

Which of the iterated integrals in Exercises below could correctly

be used to evaluate the double integral:

${\int }_1^4{\int }_2^{6}f(x,y)dydx$

Which of the iterated integrals in Exercises below could correctly

be used to evaluate the double integral:

${\int }_4^1{\int }_6^{2}f(x,y)dydx$

Which of the iterated integrals in Exercises below could correctly

be used to evaluate the double integral:

${\int }_1^4{\int }_2^{6}f(x,y)dxdy$

Which of the iterated integrals in Exercises below could correctly

be used to evaluate the double integral:

${\int }_2^6{\int }_1^{4}f(x,y)dxdy$

Which of the iterated integrals in Exercises 9–12 could correctly be used to evaluate the double integral ${\iint }_{\Omega }f(x,y)dA$

${\int }_0^2{\int }_0^{-x+2}f(x,y)dydx$

Which of the iterated integrals in Exercises 9–12 could correctly be used to evaluate the double integral ${\iint }_{\Omega }f(x,y)dA$

${\int }_0^{-x+2}{\int }_0^{2}f(x,y)dxdy$

Which of the iterated integrals in Exercises 9–12 could correctly be used to evaluate the double integral ${\iint }_{\Omega }f(x,y)dA,$

 ${\int }_0^2{\int }_0^{-y+2}f(x,y)dxdy$

Which of the iterated integrals in Exercises 9–12 could correctly be used to evaluate the double integral ${\iint }_{0}f(x,y)dA$

${\int }_2^0{\int }_{-y+2}^{0}f(x,y)dxdy$

Following region is bounded by functions $y=\frac{1}{2}x\text{ and }y=\sqrt{x}$.

Express $\Omega$ as type I or type II region. If $\Omega$ is a type I region, what are

$a, b$? If $\Omega$ is a type II region, what are $c, d$?

Explain why the double integral ${\iint }_{\Omega }dA$ gives the area of the region $\Omega$ . Illustrate your explanation with an example.

Let $g_1\left(x\right)\text{ and }{g}_2\left(x\right)$ be two continuous functions such that $g_1\left(x\right)\le g_2\left(x\right)$ on the interval $[a, b]$, and let  be the region in the $xy$-plane bounded by $g_1\text{ and }{g}_2\text{ on }[a,b]$. Use your answer to Exercise 14 to set up an iterated integral whose value is the area of $\Omega$. How is this iterated integral related

to the definite integral you would have used to compute the area of $\Omega$ in Chapter 4?

Let $h_1\left(y\right)\text{ and }{h}_2\left(y\right)$be two continuous functions such that $g_1\left(x\right)\le g_2\left(x\right)\text{ on }[a,b]$ and let $\Omega$ be region in $xy$ plane bounded by $g_1\text{ and }{g}_2\text{ on }[a,b]$. Use your answer to Exercise 14 to set up an iterated integral whose value is the area of $\Omega$. How is this iterated integral related

to the definite integral you would have used to compute the area of $\Omega$ in Chapter 4?

Use the results of Exercises 15 and 16 to find the area of the region $\Omega$ shown in Exercise 13.

Express the area of the region  between the function $f\left(x\right)=x^2$ and the $x$-axis on the interval $[-3, 3]$ as an iterated integral, integrating first with respect to $x$. Express the area of  as a sum of two iterated integrals, integrating first with respect to $y$ in each. Now evaluate your integrals.