Chapter 8

Compute the mass \(M\) along the \(x\) axis, the moment \(M_{y}\) around \(x=0,\) and the center of \(\operatorname{mass} \bar{x}=M_{y} / M\). $$ m_{1}=2 \text { at } x_{1}=1, m_{2}=4 \text { at } x_{2}=2 $$

Find the surface area when curves \(1-6\) revolve around the \(x\) axis. $$ y=\sqrt{x}, \quad 2 \leqslant x \leqslant 6 $$

Find the lengths of the curves in Problems \(1-8\). $$ y=x^{3 / 2} \text { from }(0,0) \text { to }(1,1) $$

Find where the curves in \(1-12\) intersect, draw rough graphs, and compute the area between them. $$ y=x^{2}-3 \text { and } y=1 $$

$$ y=x^{2}-2 \text { and } y=0 $$

Compute the mass \(M\) along the \(x\) axis, the moment \(M_{y}\) around \(x=0,\) and the center of \(\operatorname{mass} \bar{x}=M_{y} / M\). $$ \rho=1 \text { for }-1 \leqslant x \leqslant 3 $$

Why is \(p(x)=e^{-2 x}\) not an acceptable probability density for \(x \geqslant 0\) ? Why is \(p(x)=4 e^{-2 x}-e^{-x}\) not acceptable?

Find the lengths of the curves. $$ y=\frac{1}{3}\left(x^{2}+2\right)^{3 / 2} \text { from } x=0 \text { to } x=1 $$

Find where the curves in \(1-12\) intersect, draw rough graphs, and compute the area between them. $$ y^{2}=x \text { and } x=9 $$

Compute the mass \(M\) along the \(x\) axis, the moment \(M_{y}\) around \(x=0,\) and the center of \(\operatorname{mass} \bar{x}=M_{y} / M\). $$ \rho=x^{2} \text { for } 0 \leqslant x \leqslant 1 $$

Find the lengths of the curves. $$ y=\frac{1}{5}\left(x^{2}-2\right)^{3 / 2} \text { from } x=2 \text { to } x=4 $$

Find where the curves in \(1-12\) intersect, draw rough graphs, and compute the area between them. $$ y^{2}=x \text { and } x=y+2 $$

Find the lengths of the curves. $$ y=\frac{x^{3}}{3}+\frac{1}{4 x} \text { from } x=1 \text { to } x=3 $$

Find where the curves in \(1-12\) intersect, draw rough graphs, and compute the area between them. $$ y=x^{4}-2 x^{2} \text { and } y=2 x^{2} $$

Compute the mass \(M\) along the \(x\) axis, the moment \(M_{y}\) around \(x=0,\) and the center of \(\operatorname{mass} \bar{x}=M_{y} / M\). $$ p=\sin x \text { for } 0 \leqslant x \leqslant \pi $$

If \(p(x)=C / x^{3}\) is a probability density for \(x \geqslant 1,\) find the constant \(C\) and the probabitity that \(X \leqslant 2\).

Find the lengths of the curves. $$ y=\frac{x^{4}}{4}+\frac{1}{8 x^{2}} \text { from } x=1 \text { to } x=2 $$

Find where the curves in \(1-12\) intersect, draw rough graphs, and compute the area between them. $$ x=y^{5} \text { and } y=x^{4} $$

If you choose \(x\) completely at random between 0 and \(\pi\) what is the density \(p(x)\) and the cumulative density \(F(x)\) ?

Find the lengths of the curves. $$ y=\frac{2}{3} x^{3 / 2}-\frac{1}{2} x^{1 / 2} \text { from } x=1 \text { to } x=4 $$

Find where the curves in \(1-12\) intersect, draw rough graphs, and compute the area between them. $$ y=x^{2} \text { and } y=-x^{2}+18 x $$

Find the lengths of the curves. $$ y=x^{2} \text { from }(0,0) \text { to }(1,1) $$

Find the mass \(M\), the moments \(M_{y}\) and \(M_{x}\), and the center of \(\operatorname{mass}(\bar{x}, \bar{y})\). $$ \rho=7 \text { in the square } 0 \leqslant x \leqslant 1,0 \leqslant y \leqslant 1 $$

Find the mean value \(\mu=\Sigma n p_{n}\) or \(\mu=\int x p(x) d x\). $$ p_{1}=1 / 7, p_{2}=1 / 7, \ldots, p_{7}=1 / 7 $$

The curve given by \(x=\cos ^{3} t, y=\sin ^{3} t\) is an astroid (ahypocycloid). Its non-parametric form is \(x^{2 / 3}+y^{2 / 3}=1\). Sketch the curve from \(t=0\) to \(t=\pi / 2\) and find its length.

Find where the curves in \(1-12\) intersect, draw rough graphs, and compute the area between them. $$ y=\cos x \text { and } y=\cos ^{2} x $$

Find the length from \(t=0\) to \(t=\pi\) of the curve given by \(x=\cos t+\sin t, y=\cos t-\sin t .\) Show that the curve is a circle (of what radius?).

Find where the curves in \(1-12\) intersect, draw rough graphs, and compute the area between them. $$ y=\sin \pi x \text { and } y=2 x \text { and } x=0 $$

Find the area \(M\) and the centroid \((\bar{x}, \bar{y})\) inside curves \(11-16\). $$ y=\sqrt{1-x^{2}}, y=0, x=0 \quad \text { (quarter-circle) } $$

Find the mean value \(\mu=\Sigma n p_{n}\) or \(\mu=\int x p(x) d x\). $$ p(x)=2 / \pi\left(1+x^{2}\right), \quad x \geqslant 0 $$

Find the length from \(t=0\) to \(t=\pi / 2\) of the curve given by \(x=\cos t, y=t-\sin t\).

Find where the curves in \(1-12\) intersect, draw rough graphs, and compute the area between them. $$ y=e^{x} \text { and } y=e^{2 x-1} \text { and } x=0 $$

What integral gives the length of Archimedes' spiral \(x=t \cos t, y=t \sin t ?\)

Find where the curves in \(1-12\) intersect, draw rough graphs, and compute the area between them. $$ y=e \text { and } y=e^{x} \text { and } y=e^{-x} $$

Find the distance traveled in the first second \((\) to \(t=1)\) if \(x=\frac{1}{2} t^{2}, y=\frac{1}{3}(2 t+1)^{3 / 2}\).

Find the area inside the three lines \(y=4-x, y=3 x,\) and \(y=x .\)

Find the arca bounded by \(y=12-x, y=\sqrt{x},\) and \(y=1\).

Find the arc length in \(15-18\) by numerical integration. One arch of \(y=\sin x,\) from \(x=0\) to \(x=\pi\).

Does the parabola \(y=1-x^{2}\) out to \(x=1\) sit inside or outside the unit circle \(x^{2}+y^{2}=1\) ? Find the area of the "skin" between them.

Find the arc length by numerical integration. \(y=e^{x}\) from \(x=0\) to \(x=1\).

The waiting time for a bus has probability density \((1 / 10) e^{-x / 10},\) with \(\mu=10\) minutes. What is the probability of waiting longer than 10 minutes?

The base of a lamp is constructed by revolving the quarter-circle \(y=\sqrt{2 x-x^{2}} \mid x=1\) to \(\left.x=2\right)\) around the \(y\) axis. Draw the quarter-circle, find the area integral, and compute the area.

Rotate the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1\) around the \(x\) axis to find the volume of a football. What is the volume around the \(y\) axis? If \(a=2\) and \(b=1\), locate a point \((x, y, z)\) that is in one football but not the other.

What is the volume of the loaf of bread which comes from rotating \(y=\sin x(0 \leqslant x \leqslant \pi)\) around the \(x\) axis?

What is the volume of the flying saucer that comes from rotating \(y=\sin x(0 \leqslant x \leqslant \pi)\) around the \(y\) axis?

(a) A fair coin comes up heads 10 times in a row. Will heads or tails be more likely on the next toss? (b) The fraction of heads after \(N\) tosses is \(x\). The expected fraction after \(2 \mathrm{~N}\) tosses is __________ .

Explain why the surface area is infinite when \(y=1 / x\) is rotated around the \(x\) axis \((1 \leqslant x<\alpha)\). But the volume of "Gabriel's horn" is It can't hold enough paint to paint its surface.

Find the speed \(d s / d t\) on the circle \(x=2 \cos 3 t, y=2 \sin 3 t\).

Draw the region bounded by the curves in \(21-28 .\) Find the volume when the region is rotated (a) around the \(x\) axis (b) around the \(y\) axis. $$ y-e^{x}=1, x=1, y=0, x=0 $$

(With thanks to Trivial Pursuit) In what state is the center of gravity of the United States - the "geographical center" or centroid?