Q. 70 - Calculus

Question

Set up and solve a definite integral to find the exact area of each surface of revolution obtained by revolving the curve $y=f\left(x\right)$ around the x-axis on the interval [a, b].

$f\left(x\right)=x^2,[a,b]=[-1,1]$

Step-by-Step Solution

Answer

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Step 1. Given Information

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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.