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

검증된 단계별 안내

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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주요 개념

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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.

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).

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.

21–30. {Use of Tech} Arc length by calculatora. 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