Problem 57
Astroid Find the area of the surface formed by revolving the portion in the first quadrant of the graph of \(x^{2 / 3}+y^{2 / 3}=4\) \(0 \leq y \leq 8\) about the \(y\) -axis.
Problem 58
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 \(x^{2}+(y-3)^{2}=4\) about the \(x\) -axis.
Problem 58
Let \(V_{1}\) and \(V_{2}\) be the volumes of the solids that result when the plane region bounded by \(y=1 / x, y=0, x=\frac{1}{4},\) and \(x=c\left(c>\frac{1}{4}\right)\) is revolved about the \(x\) -axis and \(y\) -axis, respectively. Find the value of \(c\) for which \(V_{1}=V_{2}\)
Problem 59
In Exercises 59 and 60 , set up and evaluate the definite integral that gives the area of the region bounded by the graph of the function and the tangent line to the graph at the given point. $$ f(x)=\frac{1}{x^{2}+1}, \quad\left(1, \frac{1}{2}\right) $$
Problem 59
Suspension Bridge A cable for a suspension bridge has the shape of a parabola with equation \(y=k x^{2} .\) Let \(h\) represent the height of the cable from its lowest point to its highest point and let \(2 w\) represent the total span of the bridge (see figure). Show that the length \(C\) of the cable is given by \(C=2 \int_{0}^{w} \sqrt{1+\frac{4 h^{2}}{w^{4}} x^{2}} d x\)
Problem 59
In Exercises \(57-60\), use the Theorem of Pappus to find the volume of the solid of revolution. The solid formed by revolving the region bounded by the graphs of \(y=x, y=4,\) and \(x=0\) about the \(x\) -axis
Problem 60
Set up and evaluate the definite integral that gives the area of the region bounded by the graph of the function and the tangent line to the graph at the given point. $$ y=x^{3}-2 x, \quad(-1,1) $$
Problem 60
The solid formed by revolving the region bounded by the graphs of \(y=x, y=4,\) and \(x=0\) about the \(x\) -axis The solid formed by revolving the region bounded by the graphs of \(y=2 \sqrt{x-2}, y=0,\) and \(x=6\) about the \(y\) -axis
Problem 61
The graphs of \(y=x^{4}-2 x^{2}+1\) and \(y=1-x^{2}\) intersect at three points. However, the area between the curves can be found by a single integral. Explain why this is so, and write an integral for this area.
Problem 61
In Exercises 61 and \(62,\) use the Second Theorem of Pappus, which is stated as follows. If a segment of a plane curve \(C\) is revolved about an axis that does not intersect the curve (except possibly at its endpoints), the area \(S\) of the resulting surface of revolution is given by the product of the length of \(C\) times the distance \(d\) traveled by the centroid of \(C\). A sphere is formed by revolving the graph of \(y=\sqrt{r^{2}-x^{2}}\) about the \(x\) -axis. Use the formula for surface area, \(S=4 \pi r^{2},\) to find the centroid of the semicircle \(y=\sqrt{r^{2}-x^{2}}\)
