Double and Triple Integrals

$\rho (x,y)$Identify the quantities determined by the integral expressions in Exercises 3-11. If  and $y$ are both measured in centimeters and  is a density function in grams per square centimeter, give the units of the expression.

Identify the quantities determined by the integral expressions in Exercises 3-11. If x and y are both measured in centimeters and $\rho (x,y)$ is a density function in grams per square centimeter, give the units of the expression.

${\iint }_{\Omega }x\rho (x,y)dA\text{ and }{\iint }_{\Omega }y\rho (x,y)dA$

Identify the quantities determined by the integral expressions in Exercises 3-11. If x and y are both measured in centimeters and $\rho (x,y)$ is a densily funclion in grams per square cenlimeter, give the units of the expression.

$\frac{{\iint }_{\Omega }x\rho (x,y)dA}{{\iint }_{\Omega }\rho (x,y)dA}\text{ and }\frac{{\iint }_{\Omega }y\rho (x,y)dA}{{\iint }_{\Omega }\rho (x,y)dA}$

Identify the quantities determined by the integral expressions in Exercises 3-11. If x and y are both measured in centimeters and $\rho (x,y)$ is a density function in grams per square centimeter, give the units of the expression.

${\iint }_{\Omega }{x}^2\rho (x,y)dA\text{ and }{\iint }_{\Omega }{y}^2\rho (x,y)dA$

Identify the quantities determined by the integral expressions in Exercises 3-11. If x and y are both measured in centimeters and $\rho (x,y)$ is a density function in grams per square centimeter, give the units of the expression.

${\iint }_{\Omega }\left(x^2+y^2\right)\rho (x,y)dA$

Show that when the density of the region is constant, the first moment about the $y$-axis is

$M_y={\int }_1^2{\int }_{-x+2}^{2x-1}xdydx=\frac{5}{2}$

Show that when the density of the region is constant, the first moment about the $x$-axis is

$M_x={\int }_1^2{\int }_{-x+2}^{2x-1}ydydx=2$

Explain why the location of the centroid relates only to the geometry of the region and not its mass.

Find the moments of inertia about the x - and y-axes for the semicircular lamina described in Example 2. Assume that the density at every point is proportional to the distance of the point from the origin.

In Exercises $24-30$, let $\mathcal{T}$ be the triangular region with vertices $(0,0),(1,1)$,and $(1,-1)$.

Find the centroid of $\mathcal{T}$.

If the density at each point in $\mathcal{T}$ is proportional to the point's distance from the y-axis, find the mass of$\mathcal{T}.$

Each of the integrals or integral expressions in Exercises 26 represents the area of a region in the plane. Use polar coordinates to sketch the region and evaluate the expressions 

${\int }_0^{2\pi }{\int }_0^{2+\sin 4\theta }rdrd\theta$

If the density at each point in $\mathcal{T}$ is proportional to the point's distance from the x-axis, find the mass of $\mathcal{T}$.

Each of the integrals or integral expressions in Exercises 27 represents the area of a region in the plane. Use polar coordinates to sketch the region and evaluate the expressions.

$2{\int }_{-\pi /4}^{\pi /2}{\int }_0^{(\sqrt{2}/2)+\sin \theta }rdrd\theta -2{\int }_{-\pi /2}^{-\pi /4}{\int }_0^{(\sqrt{2}/2)+\sin \theta }rdrd\theta$

Each of the integrals or integral expressions in Exercises 28 represents the area of a region in the plane. Use polar coordinates to sketch the region and evaluate the expressions.

$2{\int }_0^{2\pi /3}{\int }_0^{(1/2)+\cos \theta }rdrd\theta -2{\int }_{\pi }^{4\pi /3}{\int }_0^{(1/2)+\cos \theta }rdrd\theta$

If the density at each point in $\mathcal{T}$ is proportional to the point's distance from the $x$-axis, find the center of mass of $\mathcal{T}$.

In Exercises 29–38, find an iterated integral in polar coordinates that represents the area of the given region in the polar plane and then evaluate the integral. 

29. The region enclosed by the spiral $r=\theta$ and the $x$-axis on the interval $0\le \theta \le \pi$.

Each of the integrals or integral expressions in Exercises 39–46 represents the volume of a solid in ${\mathrm{ℝ}}^3$. Use polar coordinates to describe the solid, and evaluate the expressions 

$2{\int }_{-\pi /4}^{\pi /2}{\int }_0^{(\sqrt{2}/2)+\sin \theta }rdrd\theta -2{\int }_{-\pi /2}^{-\pi /4}{\int }_0^{(\sqrt{2}/2)+\sin \theta }rdrd\theta$

 Use a double integral to prove that the area of the circle with radius R and equation r=2$R\cos \theta is\pi R^2.$

Use a double integral to prove that the area of the circle with radius R and equation$r=2R\cos \theta is\pi R^2.$

Use a double integral with polar coordinates to prove that the area of a sector with central angle $\varphi$ in a circle of radius R is given by $A=\frac{1}{2}\varphi R^2.$

Use a double integral with polar coordinates to prove that the combined area enclosed by all of the petals of the polar rose $r=\sin (2n+1)\theta$ is the same for every positive integer $n$.

$$\begin{gathered} \mathrm{Express} \mathrm{the} \mathrm{sum} 3{\mathrm{e}}^4 + 3{\mathrm{e}}^9 + 3{\mathrm{e}}^{16} + 4{\mathrm{e}}^4 + 4{\mathrm{e}}^9 + 4{\mathrm{e}}^{16} \\ \mathrm{using} \mathrm{double}-\mathrm{summation} \mathrm{notation}. \end{gathered}$$

$$\begin{gathered} \mathrm{Express} \mathrm{the} \mathrm{sum} \frac{2}{5} + \frac{2}{6} + \frac{2}{7} + \frac{2}{8} + \frac{4}{5} + \frac{4}{6} + \frac{4}{7} + \frac{4}{8} \\ \mathrm{using} \mathrm{double}-\mathrm{summation} \mathrm{notation}. \end{gathered}$$

$\mathrm{How} \mathrm{many} \mathrm{summands} \mathrm{are} \mathrm{in} \sum _{\mathrm{j}=3}^{13}\sum _{\mathrm{k}=5}^{20}\frac{{\mathrm{j}}^2}{{\mathrm{e}}^{\mathrm{jk}}} ?$

$\mathrm{How} \mathrm{many} \mathrm{summands} \mathrm{are} \mathrm{in} \sum _{\mathrm{j}={\mathrm{j}}_0}^{\mathrm{m}}\sum _{\mathrm{k}={\mathrm{k}}_0}^{\mathrm{n}} {\mathrm{j}}^{\mathrm{k}} ?$

$\mathrm{How} \mathrm{many} \mathrm{summands} \mathrm{are} \mathrm{in}\sum _{\mathrm{i}=2}^{15} \sum _{\mathrm{j}=3}^{17}\sum _{\mathrm{k}=4}^{19} \frac{\mathrm{i}}{\mathrm{j}+\mathrm{k}} ?$

How many summands are in $\sum _{i=i_0}^l\sum _{ j=j_{0 }}^m\sum _{k=k_0}^n{k}^i{j}^k$ ?

Discuss the similarities and differences between the definition of the definite integral found in Chapter 4 and the definition of the double integral found in this section.

Explain how to construct a Riemann sum for a function of two variables over a rectangular region.

Explain how to construct a midpoint Riemann sum for a function of two variables over a rectangular region for which each $\left(x_j^{*},y_k^{*}\right)$ is the midpoint of the subrectangle

Refer to your answer to Exercise 10 or to Definition 13.3.

What is the difference between a double integral and an iterated integral?

State Fubini's theorem.

Explain why using an iterated integral to evaluate a double integral is often easier than using the definition of the double integral to evaluate the integral.

Explain how the Fundamental Theorem of Calculus is used in evaluating the iterated integral ${\int }_a^b{\int }_c^{d}f(x,y)dydx$ .

Explain how the Fundamental Theorem of Calculus is used in evaluating the iterated integral ${\int }_c^d{\int }_a^{b}f\left(x,y\right)dxdy$.

Earlier in this section, we showed that we could use Fubini’s theorem to evaluate the integral $\int {\int }_R{x}^{2}ydA$ and we showed that ${\int }_2^5{\int }_1^3{x}^{2}ydxdy=91$Now evaluate the double integral by evaluating the iterated integral ${\int }_1^3{\int }_2^5{x}^{2}ydydx$.

Explain why it would be difficult to evaluate the double integrals in Exercises 18 and 19 as iterated integrals.

$$\begin{gathered} \int {\int }_{\mathfrak{R}}{e}^{xy}dA \\ where \mathfrak{R}=\left\{\left(x,y\right)I1\le x\le 2 and 1\le y\le 3\right\} \end{gathered}$$

Explain why it would be difficult to evaluate the double integrals in Exercises 18 and 19 as iterated integrals.

$$\begin{gathered} \int {\int }_{\mathfrak{R}}\cos \left(xy\right)dA \\ where \mathfrak{R}=\left\{\left(x,y\right)I \frac{\mathrm{\pi }}{4}\le x\le \frac{\mathrm{\pi }}{2} and \frac{\mathrm{\pi }}{2}\le y\le \mathrm{\pi }\right\} \end{gathered}$$

Evaluate the sums in Exercises $23–28$.

$\sum _{j=1}^3\sum _{k=1}^2{j}^k$ 

Evaluate the sums in Exercises 23–28. 

$\sum _{j=1}^3\sum _{k=1}^2{k}^j$

Evaluate the sums in Exercises $23–28$. 
$\sum _{j=1}^3\sum _{k=1}^4(3j-4k)$

Evaluate the sums in Exercises $23–28$. 

Evaluate the sums in Exercises $23–28$.

$\sum _{i=1}^4\sum _{j=1}^3\sum _{k=1}^{2}i{j}^2{k}^3$ 

Evaluate the sums in Exercises $23–28$

$\sum _{i=1}^l\sum _{j=1}^m\sum _{k=1}^{m}i{j}^2{k}^3$

Use Definition $13.4$ to evaluate the double integrals in Exercises $29–32$.

${\iint }_{R}xydA$

where 

Use Definition $13.4$ to evaluate the double integrals in Exercises $29–32$.

${\iint }_x{x}^{2}ydA$

where $R=\left\{\right(x,y)\mid 1\le x\le 3\text{ and }0\le y\le 2\}$

Use Definition $13.4$ to evaluate the double integrals in Exercises $29–32$.

${\iint }_{R}x{y}^{3}dA$

where $R=\left\{\right(x,y)\mid -2\le x\le 2 \& -1\le y\le 1\}$

Use definition to evaluate the double integral

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

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

$$\begin{gathered} \int {\int }_{R}xydA \\ Where R=\left\{\right(x,y\left)\right|0\le x\le 2,1\le y\le 4\} \end{gathered}$$