Find the area of the surface generated when the given curve is revolved about the given axis.

y=1/4(e^2x+e^−2x), for −2≤x≤2; about the x-axis

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Step 1: Recall the formula for the surface area of a curve revolved about the x-axis. The formula is:

A = 2π ∫[a,b] y √(1 + (dy/dx)^2) dx

, where y is the function being revolved, and dy/dx is its derivative.

Step 2: Compute the derivative of the given function y = (1/4)(e^(2x) + e^(-2x)). Use the chain rule to find dy/dx. The derivative is:

dy/dx = (1/4)(2e^(2x) - 2e^(-2x))

Step 3: Simplify the expression for dy/dx. Factor out constants where possible:

dy/dx = (1/2)(e^(2x) - e^(-2x))

Step 4: Substitute y and dy/dx into the surface area formula. The integral becomes:

A = 2π ∫[-2,2] (1/4)(e^(2x) + e^(-2x)) √(1 + ((1/2)(e^(2x) - e^(-2x)))^2) dx

Step 5: Simplify the integrand as much as possible and set up the integral for evaluation. You may need to use algebraic manipulation and trigonometric identities to simplify further before solving the integral.

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주요 개념

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Surface Area of Revolution

The surface area of a solid of revolution is calculated using the formula A = 2π ∫[a to b] y(x) √(1 + (dy/dx)²) dx, where y(x) is the function being revolved, and dy/dx is its derivative. This formula accounts for the curve's shape and the distance from the axis of rotation, allowing us to find the total surface area generated by revolving the curve around the specified axis.

Parametric Derivatives

To find the derivative dy/dx for the function y = 1/4(e^(2x) + e^(-2x)), we apply differentiation rules. The derivative is essential for the surface area formula, as it represents the slope of the curve at any point, which affects the surface area when the curve is revolved around an axis.

Definite Integrals

Definite integrals are used to calculate the area under a curve between two points, in this case, from x = -2 to x = 2. The integral provides the accumulated value of the function over the specified interval, which is crucial for determining the total surface area generated by the revolution of the curve.

Find the area of the surface generated when the given curve is revolved about the given axis.y=1/4(e^2x+e^−2x), for −2≤x≤2; about the x-axis