Questions: Find the volume of the solid obtained by rotating the region bounded by (y=7 x^2, x=2, x=5), and (y=0), about the (x)-axis. [V=]

Solution Steps

To find the volume of the solid obtained by rotating the given region about the x-axis, we can use the disk method. The volume \( V \) is given by the integral of the area of circular disks with radius \( y = 7x^2 \) from \( x = 2 \) to \( x = 5 \). The formula for the volume is:

\[ V = \pi \int_{2}^{5} (7x^2)^2 \, dx \]

To find the volume \( V \) of the solid obtained by rotating the region bounded by \( y = 7x^2 \), \( x = 2 \), \( x = 5 \), and \( y = 0 \) about the x-axis, we use the disk method. The volume is given by the integral:

\[ V = \pi \int_{2}^{5} (7x^2)^2 \, dx \]

We simplify the integrand:

\[ (7x^2)^2 = 49x^4 \]

Thus, the volume integral becomes:

\[ V = \pi \int_{2}^{5} 49x^4 \, dx \]

Now we evaluate the integral:

\[ V = \pi \left[ \frac{49}{5} x^5 \right]_{2}^{5} \]

Calculating the definite integral:

\[ V = \pi \left( \frac{49}{5} (5^5) - \frac{49}{5} (2^5) \right) \]

Calculating \( 5^5 = 3125 \) and \( 2^5 = 32 \):

\[ V = \pi \left( \frac{49}{5} (3125 - 32) \right) = \pi \left( \frac{49}{5} \cdot 3093 \right) = \frac{151557\pi}{5} \]

The numerical value of the volume is:

\[ V \approx 95226.0716 \]

Final Answer