Chapter 11

Show that $$ \int_{1}^{\infty} \frac{1}{x^{a}} d x=\lim _{R \rightarrow \infty} \int_{1}^{R} \frac{1}{x^{a}} d x $$ is finite if \(1

Use the suggested substitutions to compute the antiderivatives. Also, use technology or look up the integrals in a table of integrals. a. \(\int \frac{1}{\sqrt{1-x^{2}}} d x \quad x=\sin z \quad\) b. \(\quad \int \frac{1}{1+x^{2}} d x \quad x=\tan z\) c. \(\int \frac{1}{\sqrt{1+x^{2}}} d x \quad x=\tan z\) d. \(\quad \int \frac{x}{\sqrt{1+x}} d x \quad x=z-1\) e. \(\int \sqrt{1-x^{2}} d x \quad x=\sin z\) f. \(\int \frac{1}{\sqrt{x^{2}-1}} d x \quad x=\sec z\) g. \(\int \frac{1}{1+\sqrt{x}} d x \quad x=z^{2}\) h. \(\int \frac{1-\sqrt{x}}{1+\sqrt{x}} d x \quad x=z^{2}\)

Show that \(\int_{0}^{1} \frac{1}{x^{a}} d x=\lim _{r \rightarrow 0+} \int_{r}^{1} \frac{1}{x^{a}} d x\) is finite if \(0

The Visible Human Project at the National Institutes of Health has provided numerous images of cross sections of the human body. They are being integrated into the medical education and research community through a program based at the University of Michigan. Shown in Exercise Figure 11.1 .2 are eight cross sections of the right side of the female brain. Your job is to estimate the volume of the brain. Assume that the sections are at \(1 \mathrm{~cm}\) separation, that the first section only shows brain membrane, and that the scale of the cross sections is \(1: 4 .\) Include only the brain and not the membrane which is apparent as white tissue. We found it useful to make a \(5 \mathrm{~mm}\) grid on clear plastic (the cover of a CD box), each square of which would be equivalent to \(4 \mathrm{~cm}^{2}\). The average human brain volume is \(1450 \mathrm{~cm}^{3}\).

Compare the regions whose areas are $$ \int_{1}^{\infty} \frac{1}{x^{2}} d x \quad \text { and } \quad \int_{0}^{1} \frac{1}{\sqrt{x}} d x $$

a. Write an integral that is the volume of the body with base the region of the \(x, y-\) plane bounded by $$ y_{1}=0.25 \sqrt{x} \sqrt[4]{2-x} \quad y_{2}=-0.25 \sqrt{x} \sqrt[4]{2-x} \quad 0 \leq x \leq 2 $$ and with each cross section perpendicular to the \(x\) -axis at \(x\) being a square with lower edge having endpoints \(\left[x, y_{2}(x), 0\right]\) and \(\left[x, y_{1}(x), 0\right]\) (see Exercise Figure 11.1.5A). (The value of the integral is \(4 \sqrt{2} / 15\) ). b. Write an integral that is the volume of the body with base the region of the \(\mathrm{x}, \mathrm{y}\) -plane bounded by $$ y_{1}=0.25 \sqrt{x} \sqrt[4]{2-x} \quad y_{2}=-0.25 \sqrt{x} \sqrt[4]{2-x} \quad 0 \leq x \leq 2 $$ and with each cross section perpendicular to the \(x\) -axis at \(x\) being an equilateral triangle with lower edge having endpoints \(\left[x, y_{2}(x), 0\right]\) and \(\left[x, y_{1}(x), 0\right]\) (see Exercise Figure \(11.1 .5 \mathrm{~B}\) ). (The value of the integral is \(\sqrt{6} / 15\) ).

Find the area of the surface generated by rotating the graph of \(y=2 \sqrt{x},\) \(0 \leq x \leq 1\) about the \(x\) -axis.

Consider the infinite horn, \(H\), obtained by rotating the graph of $$ y=1 / x, \quad 1 \leq x $$ about the \(x\) -axis. See Exercise Figure \(11.5 .6 .\) a. Show that the volume of the interior of \(H\) is \(\pi\). b. Show that the surface area of \(H\) is greater than. $$ \int_{1}^{\infty} 2 \pi \frac{1}{x} d x=\infty $$ c. \(H,\) then, has a finite volume and can be filled with paint, but has an infinite surface area and can not be painted! I enjoyed telling a very good class about this one day. The next day Mr. Jacks, a generally casual student, told me he could paint it, and he could. His grade in the course was an A. Can you paint \(H ?\)

Write as a sum of two integrals the surface area of the torus (doughnut) generated by rotating the circle \(x^{2}+(y-b)^{2}=a^{2}(0

The gamma function, \(\Gamma(n)=\int_{0}^{\infty} x^{n-1} e^{-x} d x\) is an important function in the study of statistics. a. Compute \(\Gamma(1)\). b. Use one step of integration by parts to compute \(\Gamma(2)\). c. Use one step of integration by parts and the previous step to compute \(\Gamma(3)\). d. Use one step of integration by parts to show that if \(n\) is an integer, \(\Gamma(n+1)=n \Gamma(n)\).

Find the volume of the right circular cone with base radius \(R\) and height \(H\).

The pyramid of Cheops, the largest of the Egyptian pyramids, is 146.6 meters tall with a square base of side 230.4 meters. What is its volume?

A linear life table is given by $$ L(x)=1-x / m \quad \text { for } \quad 1 \leq x \leq m $$ Find the mean and standard deviation of the life expectancy (age at death) for this life table.

Write as the difference of two integrals the volume of the torus (doughnut) obtained by rotating the region inside the circle \(x^{2}+(y-b)^{2}=a^{2}(0

Wildlife managers decide to lower the level of water in a lake of 8000 acre feet. They open the gates at the dam and release water at the rate of \(\frac{1000}{(t+1)^{2}}\) acre-feet/day where \(t\) is measured in days. Will they empty the lake?

Atmospheric density at altitude \(h\) meters is approximately \(1.225 e^{-0.000101 h}\) \(\mathrm{kg} / \mathrm{m}^{3}\) for \(0 \leq a \leq 5000\) meters. Compute the mass of air in a vertical one-square meter column between 0 and 5000 meters.

Algae is accumulating in a lake at a rate of \(e^{-0.05 t} \sin ^{2} \pi t\). The factor \(e^{-0.05 t}\) reflects declining available oxygen and the factor \(\sin ^{2} \pi t\) reflects diurnal oscillation. Is the amount of algae produced infinite? See Exercise Figure 11.5 .11

Algae is accumulating in a lake at a rate of \(\frac{1}{1+t} \sin ^{2} \pi t .\) The factor \(\frac{1}{1+t}\) reflects declining available oxygen and the factor \(\sin ^{2} \pi t\) reflects diurnal oscillation. Is the amount of algae produced infinite? See Exercise Figure 11.5 .12

It is a fact that $$ [\arctan x]^{\prime}=\frac{1}{1+x^{2}} $$ Compute $$ \int_{0}^{\infty} \frac{1}{1+x^{2}} d x $$