Problem 52
Volume of an Ellipsoid Consider the plane region bounded by the graph of \(\left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2}=1\) where \(a>0\) and \(b>0\). Show that the volume of the ellipsoid formed when this region revolves about the \(y\) -axis is \(\frac{4 \pi a^{2} b}{3}\).
Problem 52
Think About It Consider the equation \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1 .\) (a) Use a graphing utility to graph the equation. (b) Set up the definite integral for finding the first quadrant arc length of the graph in part (a). (c) Compare the interval of integration in part (b) and the domain of the integrand. Is it possible to evaluate the definite integral? Is it possible to use Simpson's Rule to evaluate the definite integral? Explain. (You will learn how to evaluate this type of integral in Section \(6.7 .)\)
Problem 52
Find the accumulation function \(F\). Then evaluate \(F\) at each value of the independent variable and graphically show the area given by each value of \(F\). $$ F(y)=\int_{-1}^{y} 4 e^{x / 2} d x \quad \text { (a) } F(-1) \quad \text { (b) } F(0) \quad \text { (c) } F(4) $$
Problem 53
Let \(R\) be the region bounded by \(y=1 / x,\) the \(x\) -axis, \(x=1,\) and \(x=b,\) where \(b>1 .\) Let \(D\) be the solid formed when \(R\) is revolved about the \(x\) -axis. (a) Find the volume \(V\) of \(D\). (b) Write the surface area \(S\) as an integral. (c) Show that \(V\) approaches a finite limit as \(b \rightarrow \infty\). (d) Show that \(S \rightarrow \infty\) as \(b \rightarrow \infty\).
Problem 53
In Exercises \(53-56,\) use integration to find the area of the figure having the given vertices. $$ (2,-3),(4,6),(6,1) $$
Problem 54
Think About It Match each integral with the solid whose volume it represents, and give the dimensions of each solid. (a) Right circular cone (b) Torus (c) Sphere (d) Right circular cylinder (e) Ellipsoid (i) \(2 \pi \int_{0}^{r} h x d x\) (ii) \(2 \pi \int_{0}^{r} h x\left(1-\frac{x}{r}\right) d x\) (iii) \(2 \pi \int_{0}^{r} 2 x \sqrt{r^{2}-x^{2}} d x\) (iv) \(2 \pi \int_{0}^{b} 2 a x \sqrt{1-\frac{x^{2}}{b^{2}}} d x\) (v) \(2 \pi \int_{-r}^{r}(R-x)\left(2 \sqrt{r^{2}-x^{2}}\right) d x\)
Problem 55
\mathrm{\\{} I n d i v i d u a l ~ P r o j e c t ~ \(\quad\) Select a solid of revolution from everyday life. Measure the radius of the solid at a minimum of seven points along its axis. Use the data to approximate the volume of the solid and the surface area of the lateral sides of the solid.
Problem 55
Volume of a Storage Shed A storage shed has a circular base of diameter 80 feet (see figure). Starting at the center, the interior height is measured every 10 feet and recorded in the table. \begin{tabular}{|l|c|c|c|c|c|} \hline\(x\) & 0 & 10 & 20 & 30 & 40 \\ \hline Height & 50 & 45 & 40 & 20 & 0 \\ \hline \end{tabular} (a) Use Simpson's Rule to approximate the volume of the shed. (b) Note that the roof line consists of two line segments. Find the equations of the line segments and use integration to find the volume of the shed.
Problem 56
Writing Read the article "Arc Length, Area and the Arcsine Function" by Andrew M. Rockett in Mathematics Magazine. Then write a paragraph explaining how the arcsine function can be defined in terms of an arc length. (To view this article, go to the website www.matharticles.com.)
Problem 57
In Exercises \(57-60\), use the Theorem of Pappus to find the volume of the solid of revolution. The torus formed by revolving the circle \((x-5)^{2}+y^{2}=16\) about the \(y\) -axis
