Questions: Find volume by rotating a region bounded by (y=frac21+cos x) on ([0, fracpi6]) about (x)-axis.

Solution Steps

To find the volume of the solid formed by rotating the region bounded by the curve \( y = \frac{2}{1+\cos x} \) from \( x = 0 \) to \( x = \frac{\pi}{6} \) about the x-axis, we can use the method of disks or washers. The volume \( V \) is given by the integral of the area of circular disks with radius \( y \) and thickness \( dx \). The formula for the volume is:

\[ V = \pi \int_{0}^{\frac{\pi}{6}} \left(\frac{2}{1+\cos x}\right)^2 \, dx \]

To find the volume \( V \) of the solid formed by rotating the region bounded by the curve \( y = \frac{2}{1+\cos x} \) from \( x = 0 \) to \( x = \frac{\pi}{6} \) about the x-axis, we set up the volume integral using the disk method:

\[ V = \pi \int_{0}^{\frac{\pi}{6}} \left(\frac{2}{1+\cos x}\right)^2 \, dx \]

We compute the integral:

\[ V = \pi \int_{0}^{\frac{\pi}{6}} \left(\frac{2}{1+\cos x}\right)^2 \, dx \]

Calculating this integral yields a numerical value.

After evaluating the integral, we find that the volume \( V \) is approximately \( 5.026 \).

Final Answer