91Ó°ÊÓ

Consider the region between the graph of $f\left(x\right)=1-\cos x$and the x-axis on [0, π]. For each line of rotation given in Exercises 55–58, write down definite integrals that represent the

volume of the resulting solid and then use a calculator or computer to approximate the integrals.

Exercises 53-58, use Euler's method with the given $\mathrm{Δ}x$to approximate four additional points on the graph of the solution $y\left(x\right)$. Use these points to sketch a piecewise-linear approximation of the solution.

$\frac{dy}{dx}=\frac{x}{y},y\left(0\right)=2;\mathrm{Δ}x=0.5$

Consider the region between the graph of f(x) = 1 − cos x and the x-axis on [0,π]. For each line of rotation given in Exercises 55–58, write down definite integrals that represent the volume of the resulting solid and then use a calculator or computer to approximate the integrals.

In Exercises 57–62, use n frustums to approximate the area of the surface of revolution obtained by revolving the curve y = f(x) around the x-axis on the interval[a, b].

$f\left(x\right)=x^2,[a,b]=[0,4],n=2$

Use definite integrals to find the volume of each solid of revolution described in Exercises. (It is your choice whether to use disks/washers or shells in these exercises.)

The region between the graph of$f\left(x\right)=2\ln x,y=0,y=3andx=0$,revolved around the x-axis.

Consider the region between the graph of $f\left(x\right)=1-\cos x$and the $x-axis$on $\left[0,\mathrm{Ï€}\right]$. For the each line of rotation, write down definite integrals that represent the volume of the resulting solid and then use a calculator or computer to approximate the integrals.

In Exercises 57–62, use n frustums to approximate the area of the surface of revolution obtained by revolving the curve y = f(x) around the x-axis on the interval[a, b].

$f\left(x\right)=x^2,[a,b]=[0,4],n=4$

Exercises $53-58$, use Euler’s method with the given $∆x$to approximate four additional points on the graph of the solution $y\left(x\right)$. Use these points to sketch a piecewise-linear approximation of the solution.

role="math" localid="1649302672566" $\frac{dy}{dx}=x^2-y,y\left(1\right)=0;∆x=0.25$.

Use definite integrals to find the volume of each solid of revolution described in Exercises $49-61$. (It is your choice whether to use disks/washers or shells in these exercises.)

The region bounded the graph ofrole="math" localid="1651392002019" $f\left(x\right)=\frac{2}{x},y=0,y=3,x=0andx=2$, revolved around the x-axis.

Sketch slope fields for each of the differential equations in Exercises $59-64$, and within each slope field sketch four different approximate solutions of the differential equation.

$\frac{dy}{dx}=-y$.