Evaluate each of the following. (a) // « dA, where R is the region in the first quadrant bounded by the y-axis, the line y = 2, and the line y 2х. rV4-x² (b) / | (2² +y³) dy dx SI. (c) 16z dV, where E is the upper hemisphere of x2 + y² + z² = 1

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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**Evaluate each of the following:**

(a) \(\iint_R e^{y^2} \, dA\), where \(R\) is the region in the first quadrant bounded by the \(y\)-axis, the line \(y = 2\), and the line \(y = 2x\).

(b) \(\int_{0}^{2} \int_{0}^{\sqrt{4-x^2}} (x^2 + y^2) \, dy \, dx\)

(c) \(\iiint_E 16z \, dV\), where \(E\) is the upper hemisphere of \(x^2 + y^2 + z^2 = 1\).

*Graphical Explanation*:
There are no graphs or diagrams in the image. The problems involve evaluating double and triple integrals over specified regions and shapes in a coordinate system. The descriptions given define these regions algebraically.
Transcribed Image Text:**Evaluate each of the following:** (a) \(\iint_R e^{y^2} \, dA\), where \(R\) is the region in the first quadrant bounded by the \(y\)-axis, the line \(y = 2\), and the line \(y = 2x\). (b) \(\int_{0}^{2} \int_{0}^{\sqrt{4-x^2}} (x^2 + y^2) \, dy \, dx\) (c) \(\iiint_E 16z \, dV\), where \(E\) is the upper hemisphere of \(x^2 + y^2 + z^2 = 1\). *Graphical Explanation*: There are no graphs or diagrams in the image. The problems involve evaluating double and triple integrals over specified regions and shapes in a coordinate system. The descriptions given define these regions algebraically.
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