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Assume that a = 4. Integrate f(x, y) = 5y(x² + y²)³ over D using polar coordinates. (Use symbolic notation and fractions where needed.) Sy(x² + y²) ³dA =

Assume that a = 4. Integrate f(x, y) = 5y(x² + y²)³ over D using polar coordinates. (Use symbolic notation and fractions where needed.) Sy(x² + y²) ³dA =

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Assume that \( a = 4 \). Integrate \( f(x, y) = 5y(x^2 + y^2)^3 \) over \( D \) using polar coordinates. (Use symbolic notation and fractions where needed.) \[ \iint_D 5y(x^2 + y^2)^3 \, dA = \, \text{[Answer Box]} \]
Transcribed Image Text:Assume that \( a = 4 \). Integrate \( f(x, y) = 5y(x^2 + y^2)^3 \) over \( D \) using polar coordinates. (Use symbolic notation and fractions where needed.) \[ \iint_D 5y(x^2 + y^2)^3 \, dA = \, \text{[Answer Box]} \]
**Title: Identifying the Correct Region Sketch** **Problem Statement:** The region is defined as \( D: y \geq 0, \, x^2 + y^2 \leq a \). **Task:** Choose the correct sketch of the region. **Options:** 1. **First Sketch (Top-Left):** - **Description:** Half-circle with its flat side on the x-axis, extending from \(-\sqrt{a}\) to \(\sqrt{a}\) on the x-axis. The curved side lies above the x-axis. 2. **Second Sketch (Bottom-Left):** - **Description:** Another half-circle like the first, but extends from \(-a\) to \(a\) on the x-axis, with the curved side above the x-axis. 3. **Third Sketch (Top-Right):** - **Description:** A vertically oriented half-circle with its flat side on the y-axis, extending from 0 to \(a\) on the x-axis with curve spanning from \(-a\) to \(a\) on the y-axis. 4. **Fourth Sketch (Bottom-Right):** - **Description:** This sketch shows a vertically oriented half-circle with the flat side on the y-axis, extending from 0 to \(\sqrt{a}\) on the x-axis with the curve spanning from \(-\sqrt{a}\) to \(\sqrt{a}\) on the y-axis. **Guidance:** To find the correct sketch, consider the conditions \( y \geq 0 \) and \( x^2 + y^2 \leq a \). This describes a semicircle (a circle divided by the x-axis) above the x-axis.
Transcribed Image Text:**Title: Identifying the Correct Region Sketch** **Problem Statement:** The region is defined as \( D: y \geq 0, \, x^2 + y^2 \leq a \). **Task:** Choose the correct sketch of the region. **Options:** 1. **First Sketch (Top-Left):** - **Description:** Half-circle with its flat side on the x-axis, extending from \(-\sqrt{a}\) to \(\sqrt{a}\) on the x-axis. The curved side lies above the x-axis. 2. **Second Sketch (Bottom-Left):** - **Description:** Another half-circle like the first, but extends from \(-a\) to \(a\) on the x-axis, with the curved side above the x-axis. 3. **Third Sketch (Top-Right):** - **Description:** A vertically oriented half-circle with its flat side on the y-axis, extending from 0 to \(a\) on the x-axis with curve spanning from \(-a\) to \(a\) on the y-axis. 4. **Fourth Sketch (Bottom-Right):** - **Description:** This sketch shows a vertically oriented half-circle with the flat side on the y-axis, extending from 0 to \(\sqrt{a}\) on the x-axis with the curve spanning from \(-\sqrt{a}\) to \(\sqrt{a}\) on the y-axis. **Guidance:** To find the correct sketch, consider the conditions \( y \geq 0 \) and \( x^2 + y^2 \leq a \). This describes a semicircle (a circle divided by the x-axis) above the x-axis.
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