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Determine the area of the region between two curves y + y³ = x and 3y = x.

Determine the area of the region between two curves y + y³ = x and 3y = x.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem Statement:** Determine the area of the region between two curves \(y + y^3 = x\) and \(3y = x\). **Solution Approach:** To find the area between these two curves, the following steps are typically taken: 1. **Express Both Curves in Terms of One Variable**: - \( y + y^3 = x \) - \( 3y = x \) or \( y = \frac{x}{3} \) 2. **Solve for Intersection Points**: - Set \( y + y^3 \) equal to \( \frac{x}{3} \) to find the points where both curves intersect. - Solve \( y + y^3 = \frac{y}{3} \). 3. **Determine the Limits of Integration**: - Use the intersection points as the limits for integration. 4. **Integrate to Find the Area**: - Integrate the difference between the two functions over the interval defined by the intersection points. The graphs and any additional diagrams could show the region enclosed by these two curves, which then helps to set up the integral for calculating the area accurately. The graphical interpretation aids in identifying the correct limits of integration and understanding the relative positioning of each curve.
Transcribed Image Text:**Problem Statement:** Determine the area of the region between two curves \(y + y^3 = x\) and \(3y = x\). **Solution Approach:** To find the area between these two curves, the following steps are typically taken: 1. **Express Both Curves in Terms of One Variable**: - \( y + y^3 = x \) - \( 3y = x \) or \( y = \frac{x}{3} \) 2. **Solve for Intersection Points**: - Set \( y + y^3 \) equal to \( \frac{x}{3} \) to find the points where both curves intersect. - Solve \( y + y^3 = \frac{y}{3} \). 3. **Determine the Limits of Integration**: - Use the intersection points as the limits for integration. 4. **Integrate to Find the Area**: - Integrate the difference between the two functions over the interval defined by the intersection points. The graphs and any additional diagrams could show the region enclosed by these two curves, which then helps to set up the integral for calculating the area accurately. The graphical interpretation aids in identifying the correct limits of integration and understanding the relative positioning of each curve.
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