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(2) Evaluate ³ dAwhere D is the region bounded by 2y=√xand y = 5-x, yz 0 gử

(2) Evaluate ³ dAwhere D is the region bounded by 2y=√xand y = 5-x, yz 0 gử

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 2: Evaluating a Double Integral Over a Bounded Region** Evaluate the double integral \(\iint_D y^3 \, dA\) where \(D\) is the region bounded by the curves \(2y = \sqrt{x}\) and \(y = 5 - x\), with \(y \geq 0\). **Graph Explanation:** The graph provided shows the region \(D\) in the coordinate plane, shaded in gray. It is bounded by two curves: 1. **Curve \(2y = \sqrt{x}\):** - This equation expresses a relationship where \(y\) is half the square root of \(x\). - The curve is a parabola opening to the right. 2. **Line \(y = 5 - x\):** - This is a straight line with a negative slope, intersecting the y-axis at \(y = 5\) and the x-axis at \(x = 5\). The shaded region represents the area of integration for the double integral, confined by these two curves and above the line \(y = 0\). **Steps for Solution:** 1. Identify the points of intersection between the curves \(2y = \sqrt{x}\) and \(y = 5 - x\). 2. Determine the limits of integration for \(x\) and \(y\). 3. Set up the double integral with the obtained limits and evaluate it to find the solution. Ensure that the conditions \(y \geq 0\) are considered while evaluating the integral.
Transcribed Image Text:**Problem 2: Evaluating a Double Integral Over a Bounded Region** Evaluate the double integral \(\iint_D y^3 \, dA\) where \(D\) is the region bounded by the curves \(2y = \sqrt{x}\) and \(y = 5 - x\), with \(y \geq 0\). **Graph Explanation:** The graph provided shows the region \(D\) in the coordinate plane, shaded in gray. It is bounded by two curves: 1. **Curve \(2y = \sqrt{x}\):** - This equation expresses a relationship where \(y\) is half the square root of \(x\). - The curve is a parabola opening to the right. 2. **Line \(y = 5 - x\):** - This is a straight line with a negative slope, intersecting the y-axis at \(y = 5\) and the x-axis at \(x = 5\). The shaded region represents the area of integration for the double integral, confined by these two curves and above the line \(y = 0\). **Steps for Solution:** 1. Identify the points of intersection between the curves \(2y = \sqrt{x}\) and \(y = 5 - x\). 2. Determine the limits of integration for \(x\) and \(y\). 3. Set up the double integral with the obtained limits and evaluate it to find the solution. Ensure that the conditions \(y \geq 0\) are considered while evaluating the integral.
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