91Ó°ÊÓ

$$\text { Set } f(x)=\frac{\sqrt{x^{2}-9}}{x^{2}}, x \geq 3 . \text { Use a CAS }$$ (a) 10 draw the graph of \(f\) (b) to find the area between the graph of \(f\) and the \(x\) -axis from \(x=3\) to \(x=6\) (c) to locate the centroid of the region described in part (b).

Use a graphing utility to draw in one figure the graphs of both \(f(x)=1+\cos x\) and \(g(x)=\sin _{2} x\) from \(x=0\) to \(x=2 \pi\). (a) Use a CAS to find the points where the two curves is intersect; then find the area between the two curves. (b) The region between the two curves is revolved about the \(x\) -axis. Use a CAS to find the volume of the resulting solid.

The mass density of a rod that extends from \(x=2\) to \(x=3\) is given by the logarithm function \(f(x)=\ln x\). (a) Calculate the mass of the rod. (b) Find the center of mass of the rod.

(a) Use a graphing utility to draw the curve $$y^{2}=x^{2}(1-x)$$ (b) Your drawing \(\mathrm{m}\) part (a) should show that the curve forms a loop for \(0 \leq x \leq 1\). Calculate the area of the loop. HINT: Use the symmetry of the curve.

Find the volume generated by revolving the region under the graph about the \(y\) -axis. $$f(x)=\cos \frac{1}{2} \pi x, \quad x \in[0,1]$$

Find the volume generated by revolving the region under the graph about the \(y\) -axis. $$f(x)=x \sin x, \quad x \in[0, \pi]$$

Find the volume generated by revolving the region under the graph about the \(y\) -axis. $$f(x)=x e^{x}, \quad x \in[0,1]$$

Find the volume generated by revolving the region under the graph about the \(y\) -axis. $$f(x)=x \cos x, \quad x \in\left[0, \frac{1}{2} \pi\right]$$

Let \(\Omega\) be the region under the curve \(y=e^{x}, x \in[0,1] .\) Find the centroid of the solid generated by revolving \(\Omega\) about th: \(x \text { -axis. (For the appropriate formula, sce Project } 6.4 .)\)

Let \(\Omega\) be the region under the graph of \(y=\sin x, x \in\) \(\left[0, \frac{1}{2} \pi\right] .\) Find the centroid of the solid generated by revolving 2 about the \(x\) -axis. (For the appropriate formula, see Praject 6.4.)