Textbook Question
Express the area of the shaded region in Exercise 5 as the sum of two integrals with respect to y. Do not evaluate the integrals.
9. Graphical Applications of Integrals
Area Between Curves
1:15 minutesProblem 6.5.24a
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 = x³/3, for −1≤x≤1
Verified step by step guidance1
Recall the formula for the arc length of a curve defined by a function \( y = f(x) \) on the interval \( [a, b] \):
$$ L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx $$
Identify the function and the interval: here, \( y = \frac{x^3}{3} \) and \( x \) ranges from \( -1 \) to \( 1 \).
Compute the derivative \( \frac{dy}{dx} \) of the function:
$$ \frac{dy}{dx} = \frac{d}{dx} \left( \frac{x^3}{3} \right) = x^2 $$
Substitute \( \frac{dy}{dx} = x^2 \) into the arc length formula to get the integral:
$$ L = \int_{-1}^1 \sqrt{1 + (x^2)^2} \, dx = \int_{-1}^1 \sqrt{1 + x^4} \, dx $$
This integral expression represents the arc length of the curve on the given interval. It can be evaluated using a calculator or numerical methods since it does not have a simple antiderivative.
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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 = x³/3. Differentiating gives dy/dx = x², which is then squared inside the integral to find the integrand √(1 + (dy/dx)²).
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Setting up and Simplifying the Integral
After finding dy/dx, substitute it into the arc length integral and simplify the expression under the square root. For y = x³/3, the integral becomes ∫ from -1 to 1 of √(1 + x⁴) dx, which may require numerical methods or a calculator for evaluation.
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