Evaluate the following integrai. 39 4 || vx dz dx dy 0 0 v2
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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![**Problem Statement:**
Evaluate the following integral:
\[
\int_{0}^{3} \int_{0}^{9} \int_{0}^{y^2} \sqrt{x} \, dz \, dx \, dy
\]
**Explanation:**
This is a triple integral that needs to be evaluated. The limits of integration are as follows:
1. The outer integral with respect to \( y \) ranges from \( 0 \) to \( 3 \).
2. The middle integral with respect to \( x \) ranges from \( 0 \) to \( 9 \).
3. The inner integral with respect to \( z \) ranges from \( 0 \) to \( y^2 \).
The integrand is \( \sqrt{x} \). The integration starts with the innermost variable \( z \), then proceeds to \( x \), and finally to \( y \).
To solve, first integrate \(\sqrt{x}\) with respect to \(z\) over \([0, y^2]\), then integrate the resulting expression with respect to \(x\) from 0 to 9, and finally integrate with respect to \(y\) from 0 to 3.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff4f46a7d-33e6-4d1b-8c9b-a4f6142b1fc8%2F805ab983-5016-4d71-a749-7e6baf77b623%2Fie7mcc_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Evaluate the following integral:
\[
\int_{0}^{3} \int_{0}^{9} \int_{0}^{y^2} \sqrt{x} \, dz \, dx \, dy
\]
**Explanation:**
This is a triple integral that needs to be evaluated. The limits of integration are as follows:
1. The outer integral with respect to \( y \) ranges from \( 0 \) to \( 3 \).
2. The middle integral with respect to \( x \) ranges from \( 0 \) to \( 9 \).
3. The inner integral with respect to \( z \) ranges from \( 0 \) to \( y^2 \).
The integrand is \( \sqrt{x} \). The integration starts with the innermost variable \( z \), then proceeds to \( x \), and finally to \( y \).
To solve, first integrate \(\sqrt{x}\) with respect to \(z\) over \([0, y^2]\), then integrate the resulting expression with respect to \(x\) from 0 to 9, and finally integrate with respect to \(y\) from 0 to 3.
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