5.2 Double integrals over general regions  (Page 9/12)

The region $D$ is shown in the following figure. Evaluate the double integral ${\displaystyle \underset{D}{\iint }\left(x^2+y\right)dA}$ by using the easier order of integration.

The region $D$ is given in the following figure. Evaluate the double integral ${\displaystyle \underset{D}{\iint }\left(x^2-y^2\right)dA}$ by using the easier order of integration.

Find the volume of the solid under the surface $z=2x+y^2$ and above the region bounded by $y=x^5$ and $y=x.$

Find the volume of the solid under the plane $z=3x+y$ and above the region determined by $y=x^7$ and $y=x.$

Find the volume of the solid under the plane $z=x-y$ and above the region bounded by $x=\text{tan}y,x=\text{−}\text{tan}y,$ and $x=1.$

Find the volume of the solid under the surface $z=x^3$ and above the plane region bounded by $x=\text{sin}y,x=\text{−}\text{sin}y,$ and $x=1.$

Let $g$ be a positive, increasing, and differentiable function on the interval $\left[a,b\right].$ Show that the volume of the solid under the surface $z=g\prime \left(x\right)$ and above the region bounded by $y=0,$ $y=g\left(x\right),$ $x=a,$ and $x=b$ is given by $\frac{1}{2}\left(g^2\left(b\right)-g^2\left(a\right)\right).$

Let $g$ be a positive, increasing, and differentiable function on the interval $\left[a,b\right],$ and let $k$ be a positive real number. Show that the volume of the solid under the surface $z=g\prime \left(x\right)$ and above the region bounded by $y=g\left(x\right),y=g\left(x\right)+k,x=a,$ and $x=b$ is given by $k\left(g\left(b\right)-g\left(a\right)\right).$

Find the volume of the solid situated in the first octant and determined by the planes $z=2,$ $z=0,x+y=1,x=0,\text{and}y=0.$

Find the volume of the solid situated in the first octant and bounded by the planes $x+2y=1,$ $x=0,y=0,z=4,\text{and}z=0.$

Find the volume of the solid bounded by the planes $x+y=1,x-y=1,x=0,z=0,$ and $z=10.$

Find the volume of the solid bounded by the planes $x+y=1,x-y=1,x+y=\mathrm{-1},$ $x-y=\mathrm{-1},z=1\text{and}z=0.$

Let $S_1$ and $S_2$ be the solids situated in the first octant under the planes $x+y+z=1$ and $x+y+2z=1,$ respectively, and let $S$ be the solid situated between $S_1,S_2,x=0,\text{and}y=0.$

1. Find the volume of the solid $S_1.$
2. Find the volume of the solid $S_2.$
3. Find the volume of the solid $S$ by subtracting the volumes of the solids $S_1\text{and}{S}_2.$

Let $S_1\text{and}{S}_2$ be the solids situated in the first octant under the planes $2x+2y+z=2$ and $x+y+z=1,$ respectively, and let $S$ be the solid situated between $S_1,S_2,x=0,\text{and}y=0.$

1. Find the volume of the solid $S_1.$
2. Find the volume of the solid $S_2.$
3. Find the volume of the solid $S$ by subtracting the volumes of the solids $S_1\text{and}{S}_2.$

Let $S_1\text{and}{S}_2$ be the solids situated in the first octant under the plane $x+y+z=2$ and under the sphere $x^2+y^2+z^2=4,$ respectively. If the volume of the solid $S_2$ is $\frac{4\pi }{3},$ determine the volume of the solid $S$ situated between $S_1$ and $S_2$ by subtracting the volumes of these solids.

Let $S_1$ and $S_2$ be the solids situated in the first octant under the plane $x+y+z=2$ and bounded by the cylinder $x^2+y^2=4,$ respectively.

1. Find the volume of the solid $S_1.$
2. Find the volume of the solid $S_2.$
3. Find the volume of the solid $S$ situated between $S_1$ and $S_2$ by subtracting the volumes of the solids $S_1$ and $S_2.$

[T] The following figure shows the region $D$ bounded by the curves $y=\text{sin}x,$ $x=0,$ and $y=x^4.$ Use a graphing calculator or CAS to find the $x$ -coordinates of the intersection points of the curves and to determine the area of the region $D.$ Round your answers to six decimal places.

[T] The region $D$ bounded by the curves $y=\text{cos}x,x=0,\text{and}y=x^3$ is shown in the following figure. Use a graphing calculator or CAS to find the x -coordinates of the intersection points of the curves and to determine the area of the region $D.$ Round your answers to six decimal places.

Suppose that $\left(X,Y\right)$ is the outcome of an experiment that must occur in a particular region $S$ in the $xy$ -plane. In this context, the region $S$ is called the sample space of the experiment and $X\text{and}Y$ are random variables. If $D$ is a region included in $S,$ then the probability of $\left(X,Y\right)$ being in $D$ is defined as $P[\left(X,Y\right)\in D]={\displaystyle \underset{D}{\iint }p(x,y)dxdy},$ where $p(x,y)$ is the joint probability density of the experiment. Here, $p(x,y)$ is a nonnegative function for which ${\displaystyle \underset{S}{\iint }p(x,y)dxdy=1}.$ Assume that a point $\left(X,Y\right)$ is chosen arbitrarily in the square $[0,3]\times [0,3]$ with the probability density

$p(x,y)=\{\begin{array}{cc}\frac{1}{9}\hfill & (x,y)\in [0,3]\times [0,3],\hfill \\ 0\hfill & \text{otherwise}\text{.}\hfill \end{array}$

Find the probability that the point $\left(X,Y\right)$ is inside the unit square and interpret the result.

Consider $X\text{and}Y$ two random variables of probability densities $p_1(x)$ and $p_2(x),$ respectively. The random variables $X\text{and}Y$ are said to be independent if their joint density function is given by $p(x,y)=p_1(x)p_2(y).$ At a drive-thru restaurant, customers spend, on average, $3$ minutes placing their orders and an additional $5$ minutes paying for and picking up their meals. Assume that placing the order and paying for/picking up the meal are two independent events $X$ and $Y.$ If the waiting times are modeled by the exponential probability densities

$\begin{array}{ccccccc}{p}_1(x)=\{\begin{array}{cc}\frac{1}{3}{e}^{\text{−}x\text{/}3}\hfill & x\ge 0,\hfill \\ 0\hfill & \text{otherwise,}\hfill \end{array}\hfill & & & \text{and}\hfill & & & p_2(y)=\{\begin{array}{cc}\frac{1}{5}{e}^{\text{−}y\text{/}5}\hfill & y\ge 0,\hfill \\ 0\hfill & \text{otherwise,}\hfill \end{array}\hfill \end{array}$

respectively, the probability that a customer will spend less than 6 minutes in the drive-thru line is given by $P\left[X+Y\le 6\right]={\displaystyle \underset{D}{\iint }p(x,y)dxdy},$ where $D=\left\{(x,y)\left\}\right|x\ge 0,y\ge 0,x+y\le 6\right\}.$ Find $P\left[X+Y\le 6\right]$ and interpret the result.

[T] The Reuleaux triangle consists of an equilateral triangle and three regions, each of them bounded by a side of the triangle and an arc of a circle of radius s centered at the opposite vertex of the triangle. Show that the area of the Reuleaux triangle in the following figure of side length $s$ is $\frac{s^2}{2}\left(\pi -\sqrt{3}\right).$

[T] Show that the area of the lunes of Alhazen , the two blue lunes in the following figure, is the same as the area of the right triangle ABC . The outer boundaries of the lunes are semicircles of diameters $AB\text{and}AC,$ respectively, and the inner boundaries are formed by the circumcircle of the triangle $ABC.$