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Choose the correct region of integration √³x y dy dx. Assume that in each figure, the horizontal axis is the x-axis and the vertical axis is the y-axis.

Choose the correct region of integration √³x y dy dx. Assume that in each figure, the horizontal axis is the x-axis and the vertical axis is the y-axis.

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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Question
**Transcription for Educational Website:** Choose the correct region of integration \( \int_{0}^{5} \int_{\sqrt{3x}}^{x} y \, dy \, dx \). Assume that in each figure, the horizontal axis is the x-axis and the vertical axis is the y-axis. **Graphs and Diagrams:** 1. **Top Left Graph:** - A triangle above the line \( y = \sqrt{3x} \) and below the line \( y = x \). - The base is along the x-axis. 2. **Top Right Graph:** - A shaded region bounded above by the line \( y = x \), below by the line \( y = \sqrt{3x} \), extending horizontally from \( x = 0 \) to \( x = 5 \). - The area is triangular. 3. **Bottom Left Graph:** - A rectangular region extending horizontally, with a constant y-value independent of x. This does not depict the integration limits correctly. 4. **Bottom Right Graph:** - A triangle with its base along the line \( y = \sqrt{3x} \), extending up to the line \( y = x \), with integration limits from \( x = 0 \) to \( x = 5 \). The correct region of integration is illustrated by the Top Right Graph.
Transcribed Image Text:**Transcription for Educational Website:** Choose the correct region of integration \( \int_{0}^{5} \int_{\sqrt{3x}}^{x} y \, dy \, dx \). Assume that in each figure, the horizontal axis is the x-axis and the vertical axis is the y-axis. **Graphs and Diagrams:** 1. **Top Left Graph:** - A triangle above the line \( y = \sqrt{3x} \) and below the line \( y = x \). - The base is along the x-axis. 2. **Top Right Graph:** - A shaded region bounded above by the line \( y = x \), below by the line \( y = \sqrt{3x} \), extending horizontally from \( x = 0 \) to \( x = 5 \). - The area is triangular. 3. **Bottom Left Graph:** - A rectangular region extending horizontally, with a constant y-value independent of x. This does not depict the integration limits correctly. 4. **Bottom Right Graph:** - A triangle with its base along the line \( y = \sqrt{3x} \), extending up to the line \( y = x \), with integration limits from \( x = 0 \) to \( x = 5 \). The correct region of integration is illustrated by the Top Right Graph.
Evaluate \( \int_0^5 \int_{\sqrt{3x}}^x y \, dy \, dx \) by changing to polar coordinates. (Use symbolic notation and fractions where needed.) \[ \iint_D f(r, \theta) \, dr \, d\theta = \underline{\hspace{2cm}} \]
Transcribed Image Text:Evaluate \( \int_0^5 \int_{\sqrt{3x}}^x y \, dy \, dx \) by changing to polar coordinates. (Use symbolic notation and fractions where needed.) \[ \iint_D f(r, \theta) \, dr \, d\theta = \underline{\hspace{2cm}} \]
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