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

y=x^3/2−x^1/2 / 3, for 1≤x≤2; about the x-axis

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Identify the given function and the interval: The curve is given by \(y = \frac{x^{3/2} - x^{1/2}}{3}\) for \(1 \leq x \leq 2\), and it is revolved about the x-axis.

Recall the formula for the surface area of a curve revolved about the x-axis: \(S = \int_a^b 2\pi y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\), where \(y\) is the function and \(\frac{dy}{dx}\) is its derivative.

Compute the derivative \(\frac{dy}{dx}\) of the function: First, rewrite \(y\) as \(y = \frac{1}{3} x^{3/2} - \frac{1}{3} x^{1/2}\). Then, differentiate term-by-term using the power rule.

Substitute \(y\) and \(\frac{dy}{dx}\) into the surface area integral formula: \(S = \int_1^2 2\pi \left( \frac{x^{3/2} - x^{1/2}}{3} \right) \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx\).

Set up the integral for evaluation or numerical approximation as needed to find the surface area.

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주요 개념

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Surface Area of Revolution

The surface area of a solid formed by revolving a curve around an axis is found using an integral formula. For revolution about the x-axis, the formula is S = ∫ 2πy √(1 + (dy/dx)²) dx over the given interval. This calculates the total area of the curved surface generated.

Derivative of the Function

To apply the surface area formula, you need the derivative dy/dx of the given function y(x). The derivative measures the slope of the curve at each point, which is essential for computing the arc length element √(1 + (dy/dx)²) in the integral.

Integration over the Given Interval

After setting up the integral for surface area, you must evaluate it over the specified interval [1, 2]. This involves integrating the function 2πy √(1 + (dy/dx)²) with respect to x from 1 to 2 to find the exact surface area.

Find the area of the surface generated when the given curve is revolved about the given axis.y=x^3/2−x^1/2 / 3, for 1≤x≤2; about the x-axis