적분법 전체 문제

Compare the lengths of the parabola $x={y}^{2}$ and the line $x=by$ from $(0,0)\,\text{to}\,({b}^{2},b)$ as $b$ increases. What do you notice?

Solve for the length of $x={y}^{2}$ from $(0,0)\,\text{to}\,(1,1).$ Show that $x=(1/2){y}^{2}$ from $(0,0)$ to $(2,\,2)$ is twice as long. Graph both functions and explain why this is so.

[T] Which is longer between $(1,\,1)$ and $(2,1/2):$ the hyperbola $y=1/x$ or the graph of $x+2y=3?$

Explain why the surface area is infinite when $y=1/x$ is rotated around the $x\text{-axis}$ for $1\le x<\infty ,$ but the volume is finite.

The amount of work to pump the water out of a half-full cylinder is half the amount of work to pump the water out of the full cylinder.

If the force is constant, the amount of work to move an object from $x=a$ to $x=b$ is $F(b-a).$

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.

Chapter Review Exercises

True or False? Justify your answer with a proof or a counterexample.

If the half-life of $\text{seaborgium-}266$ is $360$ ms, then $k=(\ln (2))/360.$

The volume that has a base of the ellipse ${x}^{2}/4+{y}^{2}/9=1$ and cross-sections of an equilateral triangle perpendicular to the $y\text{-axis}.$ Use the method of slicing.

The region bounded by the curve $y = {x}^{2} - x$ and the x-axis from $x=1\,\text{to}\,x=4,$ rotated around the y-axis using the washer method

$x={y}^{2}$ and $x=3y$ rotated around the y-axis using the washer method

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 $y=1-x$

Find the mass of $\rho =\frac{1}{{x}^{2}+1}$ on a disk centered at the origin with radius $4.$

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 $x\in \left(-\frac{\pi }{4},\frac{\pi }{4}\right).$