Textbook Question
Find the area of the region (see figure) in two ways.
a. Using integration with respect to x.
9. Graphical Applications of Integrals
Area Between Curves
1:41 minutesProblem 6.5.27a
Textbook Question
21–30. {Use of Tech} Arc length by calculator
a. Write and simplify the integral that gives the arc length of the following curves on the given interval.
y = cos 2x, for 0 ≤ x ≤ π
Verified step by step guidance1
Recall the formula for the arc length of a curve defined by y = f(x) from x = a to x = b:
$$L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx$$
Identify the function and interval: here,
$$y = \cos 2x$$
with
$$0 \leq x \leq \pi$$
Find the derivative of y with respect to x:
$$\frac{dy}{dx} = \frac{d}{dx}(\cos 2x) = -2 \sin 2x$$
Square the derivative:
$$\left(\frac{dy}{dx}\right)^2 = (-2 \sin 2x)^2 = 4 \sin^2 2x$$
Write the integral for the arc length using the formula:
$$L = \int_{0}^{\pi} \sqrt{1 + 4 \sin^2 2x} \, dx$$
This integral represents the arc length of the curve on the given interval.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Arc Length Formula
The arc length of a curve y = f(x) from x = a to x = b is given by the integral ∫ from a to b of √(1 + (dy/dx)²) dx. This formula calculates the length of the curve by summing infinitesimal line segments along the curve.
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Derivative of the Function
To apply the arc length formula, you need the derivative dy/dx of the function y = cos(2x). Differentiating y with respect to x gives dy/dx = -2 sin(2x), which is essential for substituting into the arc length integral.
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Simplifying the Integral Expression
After finding dy/dx, substitute it into the arc length integral and simplify the expression under the square root. Simplification may involve trigonometric identities to make the integral easier to evaluate, especially when using a calculator.
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