Q. 53 - 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. Given Information

The given figure is

Step 2: Choose the integration technique

Examine the integrand to determine the best approach: basic rules, substitution, integration by parts, partial fractions, or trigonometric substitution.

Step 3: Perform the integration

Apply the chosen technique step by step, showing all substitutions and intermediate results.

Step 4: Evaluate and simplify

For definite integrals, apply the Fundamental Theorem of Calculus. For indefinite integrals, simplify and include \(+C\).

Step 5: State the final result

Write the final answer clearly.

Step 6: Conclude with the answer

The volume of the solid is