Chapter 8

Compute the area of the surface formed when \(f(x)=2 \sqrt{1-x}\) between -1 and 0 is rotated around the \(x\) -axis.

Find the arc length of \(f(x)=x^{3 / 2}\) on [0,2] .

A beam 10 meters long has density \(\sigma(x)=x^{2}\) at distance \(x\) from the left end of the beam. Find the center of mass \(\bar{x}\).

Find the average height of \(\cos x\) over the intervals \([0, \pi / 2],[-\pi / 2, \pi / 2]\), and \([0,2 \pi]\).

Verify that \(\pi \int_{0}^{1}(1+\sqrt{y})^{2}-(1-\sqrt{y})^{2} d y+\pi \int_{1}^{4}(1+\sqrt{y})^{2}-(y-1)^{2} d y=\frac{8}{3} \pi+\frac{65}{6} \pi=\) \(\frac{27}{2} \pi\)

Find the area bounded by the curves. \(y=x^{4}-x^{2}\) and \(y=x^{2}\) (the part to the right of the \(y\) -axis)

An object moves so that its velocity at time t is \(v(t)=-9.8 t+20 \mathrm{~m} / \mathrm{s}\). Describe the motion of the object between \(t=0\) and \(t=5,\) find the total distance traveled by the object during that time, and find the net distance traveled.

Find the arc length of \(f(x)=x^{2} / 8-\ln x\) on [1,2] .

A beam 10 meters long has density \(\sigma(x)=\sin (\pi x / 10)\) at distance \(x\) from the left end of the beam. Find the center of mass \(\bar{x}\).

Find the average height of \(x^{2}\) over the interval [-2,2] .

Verify that \(\int_{0}^{3} 2 \pi x\left(x+1-(x-1)^{2}\right) d x=\frac{27}{2} \pi\).

An object moves so that its velocity at time t is \(v(t)=\sin t .\) Set up and evaluate a single definite integral to compute the net distance traveled between \(t=0\) and \(t=2 \pi .\)

Compute the area of the surface formed when \(f(x)=x^{3}\) between 1 and 3 is rotated around the \(x\) -axis.

Find the arc length of \(f(x)=(1 / 3)\left(x^{2}+2\right)^{3 / 2}\) on the interval \([0, a]\).

A beam 4 meters long has density \(\sigma(x)=x^{3}\) at distance x from the left end of the beam. Find the center of mass \(\bar{x}\).

A water tank has the shape of an upright cylinder with radius \(r=1\) meter and height 10 meters. If the depth of the water is 5 meters, how much work is required to pump all the water out the top of the tank?

Find the average height of \(1 / x^{2}\) over the interval \([1, A] .\)

Verify that \(\int_{0}^{1} \pi\left(1-x^{2}\right)^{2} d x=\frac{8}{15} \pi\).

Find the area bounded by the curves. \(x=1-y^{2}\) and \(y=-x-1\)

An object moves so that its velocity at time \(t\) is \(v(t)=1+2 \sin t \mathrm{~m} / \mathrm{s} .\) Find the net distance traveled by the object between \(t=0\) and \(t=2 \pi,\) and find the total distance traveled during the same period.

Compute the area of the surface formed when \(f(x)=2+\cosh (x)\) between 0 and 1 is rotated around the \(x\) -axis.

Find the arc length of \(f(x)=\ln (\sin x)\) on the interval \([\pi / 4, \pi / 3]\).

Verify that \(\int 2 x \arccos x d x=x^{2} \arccos x-\frac{x \sqrt{1-x^{2}}}{2}+\frac{\arcsin x}{2}+C\)

Find the average height of \(\sqrt{1-x^{2}}\) over the interval \([-1,1] .\)

Verify that \(\int_{0}^{1} 2 \pi y \sqrt{1-y} d y=\frac{8}{15} \pi\).

Find the area bounded by the curves. \(x=3 y-y^{2}\) and \(x+y=3\)

Consider the function \(f(x)=(x+2)(x+1)(x-1)(x-2)\) on \([-2,2] .\) Find the total area between the curve and the \(x\) -axis (measuring all area as positive).

Consider the surface obtained by rotating the graph of \(f(x)=1 / x, x \geq 1,\) around the \(x\) axis. This surface is called Gabriel's horn or Toricelli's trumpet. Show that Gabriel's horn has infinite surface area.

An object moves with velocity \(v(t)=-t^{2}+1\) feet per second between \(t=0\) and \(t=2\). Find the average velocity and the average speed of the object between \(t=0\) and \(t=2\).

Use integration to find the volume of the solid obtained by revolving the region bounded by \(x+y=2\) and the \(x\) - and \(y\) -axes around the \(x\) -axis.

Find the area bounded by the curves. \(y=\cos (\pi x / 2)\) and \(y=1-x^{2}\) (in the first quadrant)

Consider the function \(f(x)=x^{2}-3 x+2\) on \([0,4] .\) Find the total area between the curve and the \(x\) -axis (measuring all area as positive).

Consider the circle \((x-2)^{2}+y^{2}=1 .\) Sketch the surface obtained by rotating this circle about the \(y\) -axis. (The surface is called a torus.) What is the surface area?

Find the arc length of \(f(x)=\cosh x\) on \([0, \ln 2]\)

A thin plate fills the upper half of the unit circle \(x^{2}+y^{2}=1 .\) Find the centroid.

A spring has constant \(k=10 \mathrm{~kg} / \mathrm{s}^{2} .\) How much work is done in compressing it \(1 / 10\) meter from its natural length?

The observation deck on the 102nd floor of the Empire State Building is 1,224 feet above the ground. If a steel ball is dropped from the observation deck its velocity at time \(t\) is approximately \(v(t)=-32 t\) feet per second. Find the average speed between the time it is dropped and the time it hits the ground, and find its speed when it hits the ground.

Find the volume of the solid obtained by revolving the region bounded by \(y=x-x^{2}\) and the \(x\) -axis around the \(x\) -axis.

Find the area bounded by the curves. \(y=\sin (\pi x / 3)\) and \(y=x\) (in the first quadrant)

Evaluate the three integrals: $$ A=\int_{0}^{3}\left(-x^{2}+9\right) d x \quad B=\int_{0}^{4}\left(-x^{2}+9\right) d x \quad C=\int_{4}^{3}\left(-x^{2}+9\right) d x $$ and verify that \(A=B+C\).

Set up the integral to find the arc length of \(\sin x\) on the interval \([0, \pi]\); do not evaluate the integral. If you have access to appropriate software, approximate the value of the integral.

A force of 2 Newtons will compress a spring from 1 meter (its natural length) to 0.8 meters. How much work is required to stretch the spring from 1.1 meters to 1.5 meters?

Find the volume of the solid obtained by revolving the region bounded by \(y=\sqrt{\sin x}\) between \(x=0\) and \(x=\pi / 2\), the \(y\) -axis, and the line \(y=1\) around the \(x\) -axis.

Find the area bounded by the curves. \(y=\sqrt{x}\) and \(y=x^{2}\)

Generalize the preceding result: rotate the ellipse given by \(x^{2} / a^{2}+y^{2} / b^{2}=1\) about the \(x\) -axis and find the surface area of the resulting ellipsoid. You should consider two cases, when \(a>b\) and when \(a