91Ó°ÊÓ

For each solid described in Exercises 21–24, set up volume integrals using both the shell and disk/washer methods. Which method produces an easier integral in each case, and why? Do not solve the integrals.

The region between the graph of $f\left(x\right)=x^2+1$an the x-axis on localid="1651505234403" $\left[0,1\right]$, revolved around the linelocalid="1651505239019" $x=3$.

Each of the definite integrals in Exercises 19–24 represents the volume of a solid of revolution obtained by rotating a region around either the x- or y-axis. Find this region.

$\mathrm{π}{∫}_1^5{\left(\frac{\mathrm{y}-1}{2}\right)}^2\mathrm{dy}$

Approximate the arc length of f (x) on [a, b], using n line segments and the distance formula. Include a sketch of f (x) and the line segments .

$f\left(x\right)=\sqrt{9-x^2},\left[a,b\right]=\left[-3,3\right],n=3$

Use antidifferentiation and/or separation of variables to solve the given differential equations. Your answers will involve unsolved constants.

$\frac{dy}{dx}=x^{2}y$

For each solid described in Exercises 21–24, set up volume integrals using both the shell and disk/washer methods. Which method produces an easier integral in each case, and why? Do not solve the integrals.

The region between the graph of $f\left(x\right)=x^2+1$an the x-axis on $\left[0,1\right]$, revolved around the line $y=-2$.

Each of the definite integrals in Exercises 19–24 represents the volume of a solid of revolution obtained by rotating a region around either the x- or y-axis. Find this region.

$\mathrm{π}{∫}_0^4(2^2-{\left(\sqrt{\mathrm{y}}\right)}^2)\mathrm{dy}$

The hydrostatic force exerted by a liquid of weight-density 70 pounds per cubic foot on the bottom of a cylindrical tank with height 8 feet and radius2 feet.

Approximate the arc length of f (x) on [a, b],using n line segments and the distance formula. Include a sketch of f (x) and the line segments .

$f\left(x\right)=\sqrt{9-x^2},\left[a,b\right]=\left[-3,3\right],n=6$

Use antidifferentiation and/or separation of variables to solve the given differential equations. Your answers will involve unsolved constants.

$\frac{dy}{dx}=x{y}^2$

Consider the region between $f\left(x\right)=\sqrt{x}$and the x-axis on$[0,4]$. For each line of rotation given in Exercises 25–28, use four shells based on the given rectangles to approximate the volume of the resulting solid.

Around the x-axis