the shell method or the disk/washer method to find the volume of the solid of revolution generated by revolving the region bounded by the graphs of $y=200-x^{2}, y=x^{2}$, and to the right of $x=1$ about the $y$-axis.
(Express numbers in exact form. Use symbolic notation and fractions where needed.)
Answer
Final Answer: The volume of the solid of revolution generated by revolving the region bounded by the graphs of \(y=200-x^{2}\), \(y=x^{2}\), and to the right of \(x=1\) about the y-axis is \(\boxed{9801\pi}\).
Step 1 ：Visualize the region bounded by the given equations. The region is bounded by the parabola \(y=200-x^{2}\), the parabola \(y=x^{2}\), and the line \(x=1\). The region is revolved about the y-axis to generate a solid of revolution.
Step 2 ：Since the axis of revolution is the y-axis, it would be more convenient to use the shell method. The formula for the volume of a solid of revolution using the shell method is \(V = 2\pi \int_{a}^{b} r(x)h(x) dx\), where \(r(x)\) is the radius of the shell and \(h(x)\) is the height of the shell.
Step 3 ：The radius of the shell is the distance from the y-axis to the shell, which is \(x\). The height of the shell is the difference between the upper and lower functions, which is \(200-x^{2}-x^{2}\).
Step 4 ：We need to integrate from \(x=1\) to \(x=\sqrt{100}\), which is the intersection point of the two parabolas.
Step 5 ：Calculate the integral to find the volume of the solid of revolution. The final volume is \(9801\pi\).
Step 6 ：Final Answer: The volume of the solid of revolution generated by revolving the region bounded by the graphs of \(y=200-x^{2}\), \(y=x^{2}\), and to the right of \(x=1\) about the y-axis is \(\boxed{9801\pi}\).
