Textbook Question
Determine whether the following statements are true and give an explanation or counterexample.
b. The area of the region between y=sin x and y=cos x on the interval [0,π/2] is ∫π/20(cosx−sinx)dx.
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Ch. 6 - Applications of Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 6, Problem 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
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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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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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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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Related Practice
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