Double and Triple Integrals

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

When you wish to evaluate the definite integral ${\int }_a^{b}f\left(x\right)dx$ of continuous function $f$, interval $\left[a,b\right]$ is never an impediment to using the Fundamental Theorem of Calculus. However, when you wish to evaluate the double integral ${\iint }_{\Omega }g(x,y)dA$ of a continuous function $g$ over $\Omega$, the region can make the evaluation process easier or harder. Why?

Sketch the region determined by the iterated integral and then evaluate the integral. For some of these integrals, it may be helpful to reverse the order of integration. 

${\int }_0^3{\int }_0^{\sqrt{9-y^2}}\sqrt{9-x^2}dxdy$

Evaluate the double integral over the specified region.

${\iint }_{\Omega }{x}^2{y}^{3}dA$ where $\Omega$ is region in first quadrant bounded by curves $y=x^2\text{ and }x=y^2$

Evaluate the double integral over the specified region. 

${\iint }_{\Omega }dA,$ where $\Omega$ is region in first quadrant bounded by curves $y=x^m\text{ and }y=x^n$where $m,n$are distinct positive integers.

Emmy oversees the operation of a sedimentation lagoon that was built and lined using the natural contours of the terrain. The bottom of the lagoon is the part of the surface

$z=\frac{1}{10}\left|x\right|+\frac{1}{10}\left|y\right|-3$

that lies below the $z=0$plane, where all units are in meters and $z=0$

represents the water level. What is the volume of the lagoon?

Let $f(x, y)$ be an integrable function on the rectangle $R=\left\{\right(x,y)\mid a\le x\le b\text{ and }c\le y\le d\}$ and $c\le y\le d\}$, and let $\alpha \in \mathrm{ℝ}$. Use the definition of the double integral to prove that

${\iint }_R\alpha f(x,y)dA=\alpha {\iint }_{R}f(x,y)dA.$

Let $f(x, y)$ and $g(x, y)$ be integrable functions on the rectangle $R=\left\{\right(x,y)\mid a\le x\le b\text{ and }c\le y\le d\}$. Use the definition of the double integral to prove that

${\iint }_R\left(f\right(x,y)+g(x,y\left)\right)dA={\iint }_{R}f(x,y)dA+{\iint }_{R}g(x,y)dA$

Let f(x, y) be a continuous function. Sketch each region $\Omega$ described in Exercises 25–28. Then set up one or more (if necessary) iterated integrals to compute $\int {\int }_{\Omega }f(x,y)dA$

(a) where you integrate first with respect to y and 

(b) where you integrate first with respect to x. 

$\Omega =\left\{\left(x,y\right) | |x|+|y|\le 1\right\}$

Let f(x, y) be a continuous function. Sketch each region $\Omega$ described in Exercises 25–28. Then set up one or more (if necessary) iterated integrals to compute ${\iint }_{\Omega }f(x,y)dA$,

(a) where you integrate first with respect to y and 

(b) where you integrate first with respect to x.

$\Omega =\left\{\left(x,y\right) | x^2+y^2\le 9\right\}$

In Exercises 29–34, sketch the region determined by the limits of the iterated integrals and then give another iterated integral (or a sum of iterated integrals if necessary) using the opposite order of integration. 

${\int }_1^2{\int }_{\ln x}^{e^x}f(x,y)dydx$

In Exercises 29–34, sketch the region determined by the limits of the iterated integrals and then give another iterated integral (or a sum of iterated integrals if necessary) using the opposite order of integration. 

${\int }_0^2{\int }_x^{-x+4}f(x,y)dydx$

In Exercises 29–34, sketch the region determined by the limits of the iterated integrals and then give another iterated integral (or a sum of iterated integrals if necessary) using the opposite order of integration.

${\int }_0^{\frac{\mathrm{\pi }}{2}}{\int }_0^{\sin y} f(x, y) dx dy$

In Exercises 29–34, sketch the region determined by the limits of the iterated integrals and then give another iterated integral (or a sum of iterated integrals if necessary) using the opposite order of integration  

${\int }_0^8{\int }_0^{\sqrt[3]{y}} f(x, y) dx dy$

In Exercises 29–34, sketch the region determined by the limits of the iterated integrals and then give another iterated integral (or a sum of iterated integrals if necessary) using the opposite order of integration. 

${\int }_0^{\frac{\mathrm{\pi }}{2}}{\int }_{\sin y}^{1}f(x,y)dxdy$

In Exercises 29–34, sketch the region determined by the limits of the iterated integrals and then give another iterated integral (or a sum of iterated integrals if necessary) using the opposite order of integration 

${\int }_0^8{\int }_{\sqrt[3]{y}}^{2}f(x,y)dxdy$

Find the volume of the solid bounded above by the given function over the specified region 

$f(x, y) = 10 - 2x + y$, with $\Omega$ the region from Exercise 21 

find the volume of the solid bounded above by the given function over the specified region   

$f(x, y) = 10 - 2x + y$, with $\Omega$ the region from Exercise 22. 

In Exercises 35–40, find the volume of the solid bounded above by the given function over the specified region$\Omega$. $f(x,y)=\sqrt{4-x^2-y^2}$

Region: 

In Exercises 35–40, find the volume of the solid bounded above by the given function over the specified region$\Omega =\left\{\left(x,y\right)\right| 0\le x\le \frac{\mathrm{\pi }}{4}$and $\sin x\le y\le \cos x\}$ $f(x,y)=x^{2}y.$

In Exercises 35–40, find the volume of the solid bounded above by the given function over the specified region

Find the volumes of the solids described in Exercises 41–44. The portion of the first octant bounded by the coordinate planes and the plane$3x+4y+6z=12$

Find the volumes of the solids described in Exercises 41–44. The solid bounded above by the paraboloid with equation$z=8-x^2-y^2$and bounded below by the rectangle$R=\left\{\right(x,y\left)\right|1\le x\le 2$and $0\le y\le 2\}$ in the xy-plane.

Find the volumes of the solids described in Exercises 41–44. The solid bounded above by the hyperboloid with equation$z=x^2-y^{2 }$and bounded below by the square with vertices: $(2, 2,-4), (2,-2,-4), (-2,-2,-4), (-2, 2,-4).$

Evaluate the iterated integrals in Exercises 45–48 by reversing the order of integration. Explain why it is easier to reverse the order of integration than evaluate the given iterated integral.

Evaluate the iterated integrals in Exercises 45–48 by reversing the order of integration. Explain why it is easier to reverse the order of integration than evaluate the given iterated integral.

Evaluate the iterated integrals in Exercises 45–48 by reversing the order of integration. Explain why it is easier to reverse the order of integration than evaluate the given iterated integral.

Evaluate the iterated integrals in Exercises 45–48 by reversing the order of integration. Explain why it is easier to reverse the order of integration than evaluate the given iterated integral.

${\int }_0^1{\int }_{\frac{\mathrm{\pi }}{4}}^{co{t}^{-1}y}csc x \mathrm{dx} \mathrm{dy}$

In Exercises 49–58, sketch the region determined by the iterated integral and then evaluate the integral. For some of these integrals, it may be helpful to reverse the order of integration.

In Exercises 49–58, sketch the region determined by the iterated integral and then evaluate the integral. For some of these integrals, it may be helpful to reverse the order of integration.

${\int }_0^3{\int }_{y+2}^{2y-3}\left(x^2+3xy\right) dx dy$

In Exercises 49–58, sketch the region determined by the iterated integral and then evaluate the integral. For some of these integrals, it may be helpful to reverse the order of integration.

${\int }_4^9{\int }_2^{\sqrt{x}}\left(x^3+y^2\right) dy dx$

In Exercises 49–58, sketch the region determined by the iterated integral and then evaluate the integral. For some of these integrals, it may be helpful to reverse the order of integration.

In Exercises 49–58, sketch the region determined by the iterated integral and then evaluate the integral. For some of these integrals, it may be helpful to reverse the order of integration.

In Exercises 49–58, sketch the region determined by the iterated integral and then evaluate the integral. For some of these integrals, it may be helpful to reverse the order of integration.

In Exercises 49–58, sketch the region determined by the iterated integral and then evaluate the integral. For some of these integrals, it may be helpful to reverse the order of integration.

In Exercises 49–58, sketch the region determined by the iterated integral and then evaluate the integral. For some of these integrals, it may be helpful to reverse the order of integration.

Sketch the region determined by the iterated integral and then evaluate the integral. For some of these integrals, it may be helpful to reverse the order of integration.

${\int }_{\pi /4}^{3\pi /4}{\int }_0^{\csc y}\csc ydxdy$

In Exercises 59–62, evaluate the double integral over the specified region.

$\int {\int }_{\Omega }x{e}^{x^3}dA$

Evaluate the double integral over the specified region.

${\iint }_{\Omega }x{e}^{x^3}dA$ where $\Omega$is the triangular region in the first quadrant bounded below by the x-axis, bounded above $y=m{x}_{,}\text{where }m>0$and bounded on the

right by the line with equation $x=1$.

Leila is designing a new summer range management unit for caribou in the Selkirk Mountains in the Idaho panhandle. The old unit was laid out as a rectangle, which

had nothing to do with the behavior of the caribou. The new one is supposed to resemble the actual area in which the caribou live. Leila has used a study which indicates that the density of caribou in this region in July is approximated by 

$d(x,y)=0.08x^2{y}^2-0.456x^{2}y-0.08x^2-$

$0.328x{y}^2+1.87xy+0.328x-0.061y^2+0.347y+0.061$.

Her proposed southern boundary for the management unit is a mountain ridge that roughly follows the curve $0.0195x^4$ while the northern border is a political boundary at $\frac{x}{4}+4$. The western boundary is a state line on which she places

the $y$-axis. Roughly how many caribou can be found in the management unit?

Prove Theorem 13.10 (a). That is, show that if $f(x, y)$ is an integrable function on the general region  and $c \in R$, then

${\iint }_{\Omega }\alpha f(x,y)dA=\alpha {\iint }_{\Omega }f(x,y)dA$

Prove Theorem 13.10 (b). That is, show that if $f(x, y)$ and $g(x, y)$ are integrable functions on the general region ,then

${\iint }_{\Omega }\left(f\right(x,y)+g(x,y\left)\right)dA={\iint }_{\Omega }f(x,y)dA+{\iint }_{\Omega }g(x,y)dA$

Let $a,b,c$ be positive real numbers. Prove that the volume of the pyramid with vertices $(0,0,0), (a,0,0), (0,b,0), (0,0,c)$ is $\frac{1}{6}abc.$

Let $a, c$be positive real numbers. Prove that the volume of the pyramid with vertices $(-a,a,0),(a,-a,0),(-a,-a,0),\text{ and }(0,0,c)\text{ is }\frac{4}{3}{a}^{2}c$. 

Let $[a_1,a_2],[b_1,b_2],[c_1,c_2]$ be three closed intervals explain why the triple integral ${\int }_{a_1}^{a_2}{\int }_{b_1}^{b_2}{\int }_{c_1}^{c_2}dzdydx$ computes the volume of the rectangular solid with length $a_2-a_1$, width $b_2-b_1$ and height $c_2-c_1$

Evaluate the triple integral ${\int }_0^2{\int }_0^{-\frac{3}{2}x+3}{\int }_0^{4-2x-\frac{4}{3}y}dzdydx$ and given physical interpretation of the integral

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

$2{\int }_0^{\mathrm{\pi }/2} {\int }_0^{\sin 3\mathrm{\theta }} \mathrm{rdrd\theta }$

Each of the integral expressions that follow represents the area of a region in the plane bounded by a function expressed in polar coordinates. Use the ideas from this section and from Chapter 9 to sketch the regions, and then evaluate each integral 

$\frac{1}{2}{\int }_0^{\mathrm{\pi }}{\cos }^{2}3\theta d\theta$

Each of the integral expressions that follow represents the area of a region in the plane bounded by a function expressed in polar coordinates. Use the ideas from this section and from Chapter 9 to sketch the regions, and then evaluate each integral  

$4{\int }_0^{\frac{\mathrm{\pi }}{4}}CO{S}^{2}2\theta d\theta$

Each of the integral expressions that follow represents the area of a region in the plane bounded by a function expressed in polar coordinates. Use the ideas from this section and from Chapter 9 to sketch the regions, and then evaluate each integral   

${\int }_0^{\frac{2\mathrm{\pi }}{3}}{\left(\frac{1}{2}+\cos \theta \right)}^{2}d\theta -{\int }_{\pi }^{\frac{4\mathrm{\pi }}{3}}{\left(\frac{1}{2}+\cos \theta \right)}^{2}d\theta$