적분법 전체 문제

The disk method can be used in any situation in which the washer method is successful at finding the volume of a solid of revolution.

For the following exercises, use the requested method to determine the volume of the solid.

$x=2{y}^{2}-{y}^{3},x=0,\,\text{and}\,y=0$ rotated around the x-axis using cylindrical shells

For the following exercises, find

a. the area of the region, b. the volume of the solid when rotated around the x-axis, and c. the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you.

$y={x}^{3},x=0,y=0,\,\text{and}\,x=2$

For the following exercises, find

a. the area of the region, b. the volume of the solid when rotated around the x-axis, and c. the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you.

$y={x}^{2}-x\,\text{and}\,x=0$

For the following exercises, find

a. the area of the region, b. the volume of the solid when rotated around the x-axis, and c. the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you.

[T] $y=\ln (x)+2\,\text{and}\,y=x$

For the following exercises, find

a. the area of the region, b. the volume of the solid when rotated around the x-axis, and c. the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you.

$y={x}^{2}$ and $y=\sqrt{x}$

For the following exercises, find

a. the area of the region, b. the volume of the solid when rotated around the x-axis, and c. the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you.

$y=5+x,$ $y={x}^{2},$ $x=0,$ and $x=1$

For the following exercises, find

a. the area of the region, b. the volume of the solid when rotated around the x-axis, and c. the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you.

Below ${x}^{2}+{y}^{2}=1$ and above $y=1-x$

For the following exercises, find

a. the area of the region, b. the volume of the solid when rotated around the x-axis, and c. the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you.

Find the center of mass for $\rho ={\tan }^{2}x$ on $x\in \left(-\frac{\pi }{4},\frac{\pi }{4}\right).$

For this exercise, consider the Karun-3 dam in Iran. Its shape can be approximated as an inverted isosceles triangle spanning across the river, with height 205 m and width (across the top of the dam) 388 m. Assume the current depth of the water is 180 m. The density of water is 1000 kg/m³. Find the total force on the wall of the dam.

You are a crime scene investigator attempting to determine the time of death of a victim. It is noon and $45^\circ \text{F}$ outside and the temperature of the body is $78^\circ \text{F}.$ You know the cooling constant is $k=0.00824^\circ \text{F/min}.$ When did the victim die, assuming that a human’s temperature is $98^\circ \text{F}$ ?

Find surface area of the catenoid $y=\cosh (x)$ from $x=-1$ to $x=1$ that is created by rotating this curve around the $x\,\text{-axis.}$

Derive the following formulas using the technique of integration by parts. Assume that n is a positive integer. These formulas are called reduction formulas because the exponent in the x term has been reduced by one in each case. The second integral is simpler than the original integral.

$\int {x}^{n}{e}^{x}dx={x}^{n}{e}^{x}-n\int {x}^{n-1}{e}^{x}dx$

Derive the following formulas using the technique of integration by parts. Assume that n is a positive integer. These formulas are called reduction formulas because the exponent in the x term has been reduced by one in each case. The second integral is simpler than the original integral.

$\int {x}^{n}\cos x\,dx={x}^{n}\sin x-n\int {x}^{n-1}\sin x\,dx$

A particle moving along a straight line has a velocity of $v(t)={t}^{2}{e}^{-t}$ after t sec. How far does it travel in the first 2 sec? (Assume the units are in feet and express the answer in exact form.)

Find the area of the region enclosed by the curve $y=x\cos x$ and the x-axis for

$\frac{11\pi }{2}\le x\le \frac{13\pi }{2}.$ (Express the answer in exact form.)

Find the volume of the solid generated by revolving the region bounded by the curve $y=\ln x,$ the x-axis, and the vertical line $x={e}^{2}$ about the x-axis. (Express the answer in exact form.)

Find the volume of the solid generated by revolving the region bounded by the curve $y=4\cos x$ and the x-axis, $\frac{\pi }{2}\le x\le \frac{3\pi }{2},$ about the x-axis. (Express the answer in exact form.)

Find the volume of the solid generated by revolving the region in the first quadrant bounded by $y={e}^{x}$ and the x-axis, from $x=0$ to $x=\ln (7),$ about the *y-*axis. (Express the answer in exact form.)

In Exercises 27–30, find the area of the triangle formed by the given three points.

$(1,1)$, $(2,3)$, and $(3,3)$