Questions: Find the volume of the solid of revolution formed by rotating about the x-axis the region bounded by the curves f(x)=4-x^2 and y=0. The volume is cubic units. (Type an exact answer, using π as needed.)

Solution Steps

To find the volume of the solid of revolution formed by rotating the region bounded by the curves \( f(x) = 4 - x^2 \) and \( y = 0 \) about the \( x \)-axis, we can use the disk method. The volume \( V \) is given by the integral:

\[ V = \pi \int_{a}^{b} [f(x)]^2 \, dx \]

where \( a \) and \( b \) are the points where the curve intersects the \( x \)-axis.

1. Determine the points of intersection of \( f(x) = 4 - x^2 \) and \( y = 0 \).
2. Set up the integral using the disk method.
3. Evaluate the integral to find the volume.

To find the points of intersection of the curves \( f(x) = 4 - x^2 \) and \( y = 0 \), we solve the equation: \[ 4 - x^2 = 0 \] This gives: \[ x^2 = 4 \] \[ x = \pm 2 \] Thus, the points of intersection are \( x = -2 \) and \( x = 2 \).

Using the disk method, the volume \( V \) of the solid of revolution is given by: \[ V = \pi \int_{-2}^{2} (4 - x^2)^2 \, dx \]

We evaluate the integral: \[ V = \pi \int_{-2}^{2} (4 - x^2)^2 \, dx \] \[ V = \pi \left[ \frac{512}{15} \right] \] \[ V = \frac{512\pi}{15} \]

The numerical value of the volume is: \[ V \approx 107.2330 \, \text{cubic units} \]

Final Answer