Q. 69 - 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].

$f\left(x\right)=e^{4x},[a,b]=[-1,1]$

Step-by-Step Solution

Answer

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Step 1: Surface area formula

The surface area of revolution of \(y = f(x)\) about the x-axis is \(S = 2\pi\int_a^b f(x)\sqrt{1+[f'(x)]^2}\,dx\).

Step 2: Set up and evaluate

Substitute the specific function and bounds, compute \(f'(x)\), and evaluate the integral.