Chapter 8

Find the arc length of \(y=e^{x}\) on the interval \([0,1] .\) (This can be done exactly; it is a bit tricky and a bit long.)

A thin plate lies in the region contained by \(y=x^{1 / 3}\) and the \(x\) -axis between \(x=0\) and \(x=1 .\) Find the centroid.

The equation \(x^{2} / 9+y^{2} / 4=1\) describes an ellipse. Find the volume of the solid obtained by rotating the ellipse around the \(x\) -axis and also around the \(y\) -axis. These solids are called ellipsoids; one is vaguely rugby-ball shaped, one is sort of flying-saucer shaped, or perhaps squished- beach-ball-shaped.

Use integration to compute the volume of a sphere of radius \(r .\) You should of course get the well-known formula \(4 \pi r^{3} / 3\).

Find the area bounded by the curves. \(y=\sin x \cos x\) and \(y=\sin x, 0 \leq x \leq \pi\)

Find the arc length of the function on the given interval. $$ f(x)=\sqrt{8} x \text { on }[-1,1] $$

A thin plate lies in the region between the circle \(x^{2}+y^{2}=4\) and the circle \(x^{2}+y^{2}=1\), above the \(x\) -axis. Find the centroid.

A hemispheric bowl of radius \(r\) contains water to a depth h. Find the volume of water in the bowl.

Find the arc length of the function on the given interval. $$ f(x)=\frac{1}{3} x^{3 / 2}-x^{1 / 2} \text { on }[0,1] $$

A thin plate lies in the region between the circle \(x^{2}+y^{2}=4\) and the circle \(x^{2}+y^{2}=1\) in the first quadrant. Find the centroid.

The base of a tetrahedron (a triangular pyramid) of height \(h\) is an equilateral triangle of side s. Its cross-sections perpendicular to an altitude are equilateral triangles. Express its volume \(V\) as an integral, and find a formula for \(V\) in terms of \(h\) and s. Verify that your answer is \((1 / 3)\) (area of base) \((\) height \() .\)

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

Find the arc length of the function on the given interval. $$ f(x)=\frac{1}{12} x^{3}+\frac{1}{x} \text { on }[1,4] $$

A thin plate lies in the region between the circle \(x^{2}+y^{2}=25\) and the circle \(x^{2}+y^{2}=16\) above the \(x\) -axis. Find the centroid.

The base of a solid is the region between \(f(x)=\cos x\) and \(g(x)=-\cos x,-\pi / 2 \leq x \leq\) \(\pi / 2\), and its cross-sections perpendicular to the \(x\) -axis are squares. Find the volume of the solid.

Find the arc length of the function on the given interval. $$ f(x)=2 x^{3 / 2}-\frac{1}{6} \sqrt{x} \text { on }[0,9] $$

Find the arc length of the function on the given interval. $$ f(x)=\cosh x \text { on }[-\ln 2, \ln 2] $$

Find the arc length of the function on the given interval. $$ f(x)=\frac{1}{2}\left(e^{x}+e^{-x}\right) \text { on }[0, \ln 5] $$

Find the arc length of the function on the given interval. $$ f(x)=\frac{1}{12} x^{5}+\frac{1}{5 x^{3}} \text { on }[.1,1] $$

Find the arc length of the function on the given interval. $$ f(x)=\ln (\sin x) \text { on }[\pi / 6, \pi / 2] $$

Find the arc length of the function on the given interval. $$ f(x)=\ln (\cos x) \text { on }[0, \pi / 4] $$

Set up the integral to compute the arc length of the function on the given interval. Do not evaluate the integral. $$ f(x)=x^{2} \text { on }[0,1] $$

Set up the integral to compute the arc length of the function on the given interval. Do not evaluate the integral. $$ f(x)=x^{10} \text { on }[0,1] $$

Set up the integral to compute the arc length of the function on the given interval. Do not evaluate the integral. $$ f(x)=\sqrt{x} \text { on }[0,1] $$

Set up the integral to compute the arc length of the function on the given interval. Do not evaluate the integral. $$ f(x)=\ln x \text { on }[1, e] $$

Set up the integral to compute the arc length of the function on the given interval. Do not evaluate the integral. \(f(x)=\sqrt{1-x^{2}}\) on \([-1,1] .\) (Note this describes the top half of a circle with radius \(1 .)\)

Set up the integral to compute the arc length of the function on the given interval. Do not evaluate the integral. \(f(x)=\sqrt{1-x^{2} / 9}\) on [-3,3] (Note: this describes the top half of an ellipse with a major axis of length 6 and a minor axis of length 2.)

Set up the integral to compute the arc length of the function on the given interval. Do not evaluate the integral. $$ f(x)=\frac{1}{x} \text { on }[1,2] $$