Q. 54 - Calculus

Question

Consider the region between the graphs of $f\left(x\right)={\left(x-2\right)}^2$and $g\left(x\right)=x$ on $\left[1,4\right]$. For each line of rotation given in Exercises 51–54, 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 \( f(x) = (x-2)^2 \) and \( g(x) = x \) on \( [1, 4] \), determine which function is larger and set up the volume integral based on the given line of rotation.

Step 2: Evaluate

Use the washer or shell method and compute the definite integral for the volume.