Use the shell method to find the volume of the solid formed when a hole of radius 5 is drilled symmetrically along the axis of a right circular cone of radius 8 and height 12. Model the situation on a set of axes by placing the center of the base of the cone at the origin and the cone's axis along the positive \( y \)-axis. Set up the integral that gives the volume of the solid using the shell method. Use nonnegative and increasing limits of integration. Select the correct choice below and fill in the answer boxes to complete your choice. (Type exact answers.) A. B.

b) \( f(x)=-2 \sqrt[4]{x} \) a. \( \frac{2}{3} x^{\frac{-1}{3}} \) b. \( \frac{2}{3} x^{2} \) c. \( -2 \cdot \frac{1}{4} x^{\frac{-3}{4}} \)

Sketch the region enclosed by \( y=e^{2 x}, y=e^{4 x} \), and \( x=1 \). Find the area of the region.

Set up the definite integral required to find the area of the region between the graph of \( y=16-x^{2} \) and \( y=-2 x-47 \) over the interval \( -3 \leq x \leq 7 \).

Realiza la investigación encome Integrar: de resolver los ejercicios: 1. \( \int x^{3} \cdot e^{x} d x \) 2. \( \int|1-x| \cdot \sqrt{x} d x \) 3. \( \int \frac{3 x}{\sqrt[3]{x^{2}+3}} d x \)

Set up the definite integral required to find the area of the region between the graph of \( y=13-x^{2} \) and \( y=24 x+148 \) over the interval \( -5 \leq x \leq 3 \).