Q. 48 - Calculus

Question

Consider the region between the graphs of $f\left(x\right)=x^2$ and $g\left(x\right)=2x$ on $\left[0,2\right]$. For each line of rotation given in Exercises 47–50, use definite integrals to find the volume of the resulting solid.

Step-by-Step Solution

Answer

The volume of the solid is

Step 1: Set up the integral

For the region between \(f(x) = x^2\) and the x-axis, the volume of revolution depends on the axis of rotation and the interval specified in the problem.

Step 2: General disk/washer method

If rotating about the x-axis over \([a,b]\): \(V = \pi\int_a^b [f(x)]^2\,dx = \pi\int_a^b x^4\,dx = \pi\left[\frac{x^5}{5}\right]_a^b\). The specific bounds determine the final answer.