Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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![**Problem Statement:**
Find the length of the curve given by \( x = \frac{y^2}{4} \), \( 0 \leq y \leq 2 \). State the integration method used to evaluate the integral.
(If applicable, you may use the integral formula for \( \sec^3 x \) derived in Trigonometric Integrals example video.)
**Instructions for Students:**
1. Identify the curve and its given equation.
2. Use the appropriate formula to find the length of the curve.
3. Clearly state the method of integration you use to solve the integral.
4. Refer to Trigonometric Integrals example video for additional guidance on using the integral formula for \( \sec^3 x \), if applicable.
**Steps to Follow:**
1. **Understand the Curve Equation:** The curve is defined by the equation \( x = \frac{y^2}{4} \).
2. **Curve Length Formula:** To find the length of the curve \( y = f(x) \) from \( y = a \) to \( y = b \), use the arc length formula:
\[
L = \int_{a}^{b} \sqrt{1 + \left( \frac{dx}{dy} \right)^2} \, dy
\]
3. **Derivative Calculation:** Compute the derivative \( \frac{dx}{dy} \):
\[
x = \frac{y^2}{4} \Rightarrow \frac{dx}{dy} = \frac{d}{dy} \left( \frac{y^2}{4} \right) = \frac{y}{2}
\]
4. **Substitute into Length Formula:** Substitute \( \frac{dx}{dy} \) into the arc length formula:
\[
L = \int_{0}^{2} \sqrt{1 + \left( \frac{y}{2} \right)^2} \, dy
\]
Simplify the integrand:
\[
L = \int_{0}^{2} \sqrt{1 + \frac{y^2}{4}} \, dy = \int_{0}^{2} \sqrt{\frac{4 + y^2}{4}} \, dy = \int_{0}^{2} \frac{\sqrt{4 + y^2}}{](https://content.bartleby.com/qna-images/question/dc22955d-cca8-418b-8553-dddb50b2093d/6ce06ffc-9fba-49f0-8805-b28f4659f4ac/wlace5i.png)
Transcribed Image Text:**Problem Statement:**
Find the length of the curve given by \( x = \frac{y^2}{4} \), \( 0 \leq y \leq 2 \). State the integration method used to evaluate the integral.
(If applicable, you may use the integral formula for \( \sec^3 x \) derived in Trigonometric Integrals example video.)
**Instructions for Students:**
1. Identify the curve and its given equation.
2. Use the appropriate formula to find the length of the curve.
3. Clearly state the method of integration you use to solve the integral.
4. Refer to Trigonometric Integrals example video for additional guidance on using the integral formula for \( \sec^3 x \), if applicable.
**Steps to Follow:**
1. **Understand the Curve Equation:** The curve is defined by the equation \( x = \frac{y^2}{4} \).
2. **Curve Length Formula:** To find the length of the curve \( y = f(x) \) from \( y = a \) to \( y = b \), use the arc length formula:
\[
L = \int_{a}^{b} \sqrt{1 + \left( \frac{dx}{dy} \right)^2} \, dy
\]
3. **Derivative Calculation:** Compute the derivative \( \frac{dx}{dy} \):
\[
x = \frac{y^2}{4} \Rightarrow \frac{dx}{dy} = \frac{d}{dy} \left( \frac{y^2}{4} \right) = \frac{y}{2}
\]
4. **Substitute into Length Formula:** Substitute \( \frac{dx}{dy} \) into the arc length formula:
\[
L = \int_{0}^{2} \sqrt{1 + \left( \frac{y}{2} \right)^2} \, dy
\]
Simplify the integrand:
\[
L = \int_{0}^{2} \sqrt{1 + \frac{y^2}{4}} \, dy = \int_{0}^{2} \sqrt{\frac{4 + y^2}{4}} \, dy = \int_{0}^{2} \frac{\sqrt{4 + y^2}}{
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