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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 6 - Applications of IntegrationProblema 6.6.25b
Capitolo 6, Problema 6.6.25b

Consider the following curves on the given intervals.  


b. Use a calculator or software to approximate the surface area.


y=cos x, for 0≤x≤π/2; about the x-axis

Guida verificata passo dopo passo
1
Identify the formula for the surface area of a solid of revolution about the x-axis: \[ S = \int_a^b 2\pi y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]
Given the curve is \(y = \cos x\) on the interval \(0 \leq x \leq \frac{\pi}{2}\), substitute \(y\) into the formula: \[ S = \int_0^{\frac{\pi}{2}} 2\pi \cos x \sqrt{1 + \left(\frac{d}{dx}(\cos x)\right)^2} \, dx \]
Compute the derivative \(\frac{dy}{dx}\): \[ \frac{d}{dx}(\cos x) = -\sin x \]
Substitute the derivative back into the integral: \[ S = \int_0^{\frac{\pi}{2}} 2\pi \cos x \sqrt{1 + (-\sin x)^2} \, dx = \int_0^{\frac{\pi}{2}} 2\pi \cos x \sqrt{1 + \sin^2 x} \, dx \]
Use a calculator or software to approximate the value of the integral, which will give the surface area of the solid formed by revolving the curve about the x-axis.

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Surface Area of a Solid of Revolution

The surface area generated by revolving a curve around an axis is found using an integral formula. For a curve y = f(x) revolved about the x-axis, the surface area is given by S = ∫ 2πy √(1 + (dy/dx)²) dx over the interval. This formula accounts for the circumference of circular slices and the curve's slope.
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Example 1: Minimizing Surface Area

Derivative of the Function y = cos x

To apply the surface area formula, the derivative dy/dx is needed. For y = cos x, the derivative is dy/dx = -sin x. This derivative measures the slope of the curve at each point, which affects the length of the surface element in the integral.
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Derivative of the Natural Exponential Function (e^x)

Numerical Approximation Using Calculators or Software

When the integral for surface area cannot be solved analytically or is complex, numerical methods like Simpson's rule or trapezoidal rule are used. Calculators or software approximate the integral value by evaluating the function at discrete points, providing an accurate estimate of the surface area.
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