Please help. This problem involves triple integrals. Thank you. ### ProblemEvaluate the integral \[\int_{0}^{25} \int_{0}^{2} \int_{4y}^{8} \frac{5 \cos(x^2)}{4\sqrt{z}} \, dx \, dy \, dz\]by changing the order of integration in an appropriate way.### ExplanationThis problem involves a triple integral with the given limits of integration. The expression to integrate is $\frac{5 \cos(x^2)}{4\sqrt{z}}$, and it is initially set up for integration with respect to $x$, $y$, and $z$ (in that order). The limits of integration are as follows:- $x$ ranges from $4y$ to $8$- $y$ ranges from $0$ to $2$- $z$ ranges from $0$ to $25$The task involves changing the order of integration so that the integral can be evaluated more easily. This may involve determining new limits of integration and re-evaluating the order in which the integrations are performed.

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Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

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Chapter2: Second-order Linear Odes

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Please help. This problem involves triple integrals. Thank you.

### Problem

Evaluate the integral

\[
\int_{0}^{25} \int_{0}^{2} \int_{4y}^{8} \frac{5 \cos(x^2)}{4\sqrt{z}} \, dx \, dy \, dz
\]

by changing the order of integration in an appropriate way.

### Explanation

This problem involves a triple integral with the given limits of integration. The expression to integrate is $\frac{5 \cos(x^2)}{4\sqrt{z}}$, and it is initially set up for integration with respect to $x$, $y$, and $z$ (in that order).

The limits of integration are as follows:

- $x$ ranges from $4y$ to $8$
- $y$ ranges from $0$ to $2$
- $z$ ranges from $0$ to $25$

The task involves changing the order of integration so that the integral can be evaluated more easily. This may involve determining new limits of integration and re-evaluating the order in which the integrations are performed.

Expert Solution

Step 1

Consider the provided question,

We have to find the integral $\int_{0}^{25} \int_{0}^{2} \int_{4 y}^{8} \frac{5 \cos (x^{2})}{4 \sqrt{z}} d x d y d z$ by changing the order of integration.

Observe that the x-limit varies from $4 y \leq x ⩽ 8$ and y limits $0 \leq y \leq 2$ .

So, changing the order of x and y, we have the limits as;

$0 \leq x \leq 8 and 0 \leq y \leq \frac{x}{4}$ .

So, the integral becomes,

$\int_{0}^{25} \int_{0}^{8} \int_{0}^{\frac{x}{4}} \frac{5 \cos (x^{2})}{4 \sqrt{z}} d y d x d z$

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