다중적분 전체 문제

The region $D$ is given in the following figure. Evaluate the double integral $\iint_{D}({x}^{2}-{y}^{2})dA$ 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 in the first octant 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 $x=\tan \,y,x=-\tan \,y,$ and $x=1.$

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

for values of $y$ between $y=\frac{-\pi }{2}\,\text{ and }\,y=\frac{\pi }{2}$

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

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

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 $x+y=1,x-y=1,x+y=-1,$ $x-y=-1,z=1\,\text{and}\,z=0.$

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 ${S}_{1},{S}_{2},x=0,\,\text{and}\,y=0.$

a. Find the volume of the solid ${S}_{1}.$ b. Find the volume of the solid ${S}_{2}.$ c. 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 ${S}_{1},{S}_{2},x=0,\,\text{and}\,y=0.$

a. Find the volume of the solid ${S}_{1}.$ b. Find the volume of the solid ${S}_{2}.$ c. 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.

Consider the plane $x + y + z = 2$ and the cylinder ${x}^{2} + {y}^{2} = 4$in the first octant. a. Find the volume under the plane. b. Find the volume inside the cylinder under the plane $z = 2$. c. Find the volume above the plane, inside the cylinder, and below the plane $z = 2$.

[T] The following figure shows the region $D$ bounded by the curves $y=\sin \,x,$ $x=0,$ 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 $y=\cos \,x,x=0,\,\text{and}\,y={x}^{3}$ 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 $(X,Y)$ 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 $(X,Y)$ being in $D$ is defined as $P[(X,Y)\in D]=\iint_{D}p(x,y)dx\,dy,$ where $p(x,y)$ is the joint probability density of the experiment. Here, $p(x,y)$ is a nonnegative function for which $\iint_{S}p(x,y)dx\,dy=1.$ Assume that a point $(X,Y)$ is chosen arbitrarily in the square $[0,3]\,\times \,[0,3]$ with the probability density

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

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

Consider $X\,\text{and}\,Y$ two random variables of probability densities ${p}_{1}(x)$ and ${p}_{2}(y),$ 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

$\begin{array}{ccccccc}{p}_{1}(x)=\left\{\begin{array}{ll}\frac{1}{3}{e}^{-x/3} & x\ge 0, \\ 0 & \text{otherwise,}\end{array}\right. & & & \text{and} & & & {p}_{2}(y)=\left\{\begin{array}{ll}\frac{1}{5}{e}^{-y/5} & y\ge 0, \\ 0 & \text{otherwise,}\end{array}\right.\end{array}$

respectively, the probability that a customer will spend less than 6 minutes in the drive-thru line is given by $P[X+Y\le 6]=\iint_{D}p(x,y)dx\,dy,$ where $D=\{(x,y)\}|x\ge 0,y\ge 0,x+y\le 6\}.$ Find $P[X+Y\le 6]$ 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 $\frac{{s}^{2}}{2}\left(\pi -\sqrt{3}\right).$