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Sketch the region bounded by the given line and curve. Then express the region's area as an iterated double integral and evaluate the integral. The parabola x = -y and the line y = x + 30

Sketch the region bounded by the given line and curve. Then express the region's area as an iterated double integral and evaluate the integral. The parabola x = -y and the line y = x + 30

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
Chapter10: Analytic Geometry
Section10.5: Equations Of Lines
Problem 50E: The y-axis along with the graphs of y=-2x+7 and y=x+2 encloses a triangular region. Find the area of...
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### Problem Statement Sketch the region bounded by the given line and curve. Then express the region's area as an iterated double integral and evaluate the integral. The parabola \(x = -\frac{y^2}{2}\) and the line \(y = x + 30\). --- ### Solution Approach 1. **Graphical Representation** - First, sketch the parabola \(x = -\frac{y^2}{2}\). This is a parabolic curve that opens to the left. - Next, sketch the straight line \(y = x + 30\). This is a linear graph that intersects the y-axis at 30 and has a slope of 1. 2. **Identifying the Region** - Determine the points of intersection between the parabola and the line. This requires solving the equations simultaneously. - Use these intersection points to define the limits of integration. 3. **Setting up the Double Integral** - Express the area of the region as a double integral. - Integrate first with respect to y, then with respect to x, over the appropriate limits derived from the intersection points. 4. **Evaluating the Integral** - Use techniques of integration to find the value of the integral, which gives the area of the region bounded by the curve and the line. ### Interpretation of Graphs and Diagrams Since the image does not contain any pre-drawn graphs or diagrams, here’s a detailed method to sketch and solve the problem: 1. **Sketching the Parabola:** - Plot the vertex of the parabola at the origin (0, 0). - Since the equation is \(x = -\frac{y^2}{2}\), the parabola opens to the left. 2. **Sketching the Line:** - The line \(y = x + 30\) intersects the y-axis at (0, 30) and has a slope of 1, indicating a 45-degree angle with positive direction along the x-axis. 3. **Finding Intersection Points** - Solve \(y = x + 30\) and \(x = -\frac{y^2}{2}\) to find the points where the curves intersect. By practicing this approach with visual aid and algebraic verification, students will have a comprehensive understanding of handling such problems involving bounded regions and double integrals.
Transcribed Image Text:### Problem Statement Sketch the region bounded by the given line and curve. Then express the region's area as an iterated double integral and evaluate the integral. The parabola \(x = -\frac{y^2}{2}\) and the line \(y = x + 30\). --- ### Solution Approach 1. **Graphical Representation** - First, sketch the parabola \(x = -\frac{y^2}{2}\). This is a parabolic curve that opens to the left. - Next, sketch the straight line \(y = x + 30\). This is a linear graph that intersects the y-axis at 30 and has a slope of 1. 2. **Identifying the Region** - Determine the points of intersection between the parabola and the line. This requires solving the equations simultaneously. - Use these intersection points to define the limits of integration. 3. **Setting up the Double Integral** - Express the area of the region as a double integral. - Integrate first with respect to y, then with respect to x, over the appropriate limits derived from the intersection points. 4. **Evaluating the Integral** - Use techniques of integration to find the value of the integral, which gives the area of the region bounded by the curve and the line. ### Interpretation of Graphs and Diagrams Since the image does not contain any pre-drawn graphs or diagrams, here’s a detailed method to sketch and solve the problem: 1. **Sketching the Parabola:** - Plot the vertex of the parabola at the origin (0, 0). - Since the equation is \(x = -\frac{y^2}{2}\), the parabola opens to the left. 2. **Sketching the Line:** - The line \(y = x + 30\) intersects the y-axis at (0, 30) and has a slope of 1, indicating a 45-degree angle with positive direction along the x-axis. 3. **Finding Intersection Points** - Solve \(y = x + 30\) and \(x = -\frac{y^2}{2}\) to find the points where the curves intersect. By practicing this approach with visual aid and algebraic verification, students will have a comprehensive understanding of handling such problems involving bounded regions and double integrals.
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