Suppose you want to find y² + x² = 9, z = : 0, and x - N 3.0 2.5 2.0 1.5 1.0 0.5 0.0 -3 -2 -1 <2<
Suppose you want to find y² + x² = 9, z = : 0, and x - N 3.0 2.5 2.0 1.5 1.0 0.5 0.0 -3 -2 -1 <2<
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Suppose you want to find \(\iiint_B f(x, y, z) \, dV\), where \(B\) is the solid in the first octant bounded by \(y^2 + z^2 = 9\), \(z = 0\), and \(x = 5\). (see picture below).
The image shows a 3D plot of a quarter-cylinder solid with its axis along the x-axis. It is bounded by the plane \(z = 0\) at the bottom, the plane \(x = 5\) at the front, and the curved surface defined by the equation \(y^2 + z^2 = 9\).
The plot is a surface extending from the y-axis to \(y = 3\), and from the z-axis to \(z = 3\), and along the x-axis up to \(x = 5\).
This can be written as a triple integral:
\[
\int_{x_1}^{x_2} \int_{y_1}^{y_2} \int_{z_1}^{z_2} f(x, y, z) \, dz \, dy \, dx,
\]
where
\[
0 \le z < \sqrt{9 - y^2}
\]
\[
0 \le y \le 3
\]
\[
0 \le x \le 5
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F09d105f3-6c69-4cbc-8997-6988f1733e6f%2F890e7f22-ecc4-431c-9eea-1ecc592e8bda%2Fy5a1rqwq_processed.png&w=3840&q=75)
Transcribed Image Text:Suppose you want to find \(\iiint_B f(x, y, z) \, dV\), where \(B\) is the solid in the first octant bounded by \(y^2 + z^2 = 9\), \(z = 0\), and \(x = 5\). (see picture below).
The image shows a 3D plot of a quarter-cylinder solid with its axis along the x-axis. It is bounded by the plane \(z = 0\) at the bottom, the plane \(x = 5\) at the front, and the curved surface defined by the equation \(y^2 + z^2 = 9\).
The plot is a surface extending from the y-axis to \(y = 3\), and from the z-axis to \(z = 3\), and along the x-axis up to \(x = 5\).
This can be written as a triple integral:
\[
\int_{x_1}^{x_2} \int_{y_1}^{y_2} \int_{z_1}^{z_2} f(x, y, z) \, dz \, dy \, dx,
\]
where
\[
0 \le z < \sqrt{9 - y^2}
\]
\[
0 \le y \le 3
\]
\[
0 \le x \le 5
\]
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