I need help with #4 **Problem 4****Given $ f(x, y) = \cos(x^2) $:**a) Draw the region that is the base (or footprint) of the surface.b) Setup BOTH integrals $\int \int \, dydx$ and $\int \int \, dxdy$.c) Then solve ONE of the integrals to find the volume of $ f(x, y) = y \cos(x^2) $ over the region defined by $ x = 1 $, $ y = 0 $, and $ y = x $.---**Explanation for the Educational Website:**This problem involves calculus and the evaluation of double integrals to determine the volume under a surface described by the function $ f(x, y) = y\cos(x^2) $. The task involves three steps:1. **Drawing the Region:** - The base region is determined by the boundaries $ x = 1 $, $ y = 0 $, and $ y = x $. - This region is a triangular section on the xy-plane with vertices at $(0,0)$, $(1,0)$, and $(1,1)$.2. **Setting Up Integrals:** - **$\int \int \, dydx:$** The order 'dydx' implies integrating with respect to $ y $ first and then $ x $. Here, $ y $ ranges from 0 to $ x $, and $ x $ ranges from 0 to 1. - **$\int \int \, dxdy:$** The order 'dxdy' implies integrating with respect to $ x $ first and then $ y $. In this configuration, $ x $ ranges from $ y $ to 1, and $ y $ ranges from 0 to 1.3. **Solving the Integral:** - Choose one of the integral setups to evaluate and find the volume of the space under the surface described by $ f(x, y) $ over the mentioned triangular region.

Answered: Image /qna-images/answer/5d95f713-3763-4416-9da7-4c103df8f351.jpg

4. Given f(x, y) = cos(x²). a) Draw the region that is the base (or footprint) the surface, b) Setup BOTH dydx and dxdy. c) Then solve ONE of the integrals to find the volume of f(x, y) = ycos(x²) over the region defined by x = 1, y = 0 and y = x.

4. Given f(x, y) = cos(x²). a) Draw the region that is the base (or footprint) the surface, b) Setup BOTH dydx and dxdy. c) Then solve ONE of the integrals to find the volume of f(x, y) = ycos(x²) over the region defined by x = 1, y = 0 and y = x.

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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Question

I need help with #4

**Problem 4**

**Given $ f(x, y) = \cos(x^2) $:**

a) Draw the region that is the base (or footprint) of the surface.

b) Setup BOTH integrals $\int \int \, dydx$ and $\int \int \, dxdy$.

c) Then solve ONE of the integrals to find the volume of $ f(x, y) = y \cos(x^2) $ over the region defined by $ x = 1 $, $ y = 0 $, and $ y = x $.

---

**Explanation for the Educational Website:**

This problem involves calculus and the evaluation of double integrals to determine the volume under a surface described by the function $ f(x, y) = y\cos(x^2) $. The task involves three steps:

1. **Drawing the Region:**
- The base region is determined by the boundaries $ x = 1 $, $ y = 0 $, and $ y = x $.
- This region is a triangular section on the xy-plane with vertices at $(0,0)$, $(1,0)$, and $(1,1)$.

2. **Setting Up Integrals:**
- **$\int \int \, dydx:$** The order 'dydx' implies integrating with respect to $ y $ first and then $ x $. Here, $ y $ ranges from 0 to $ x $, and $ x $ ranges from 0 to 1.
- **$\int \int \, dxdy:$** The order 'dxdy' implies integrating with respect to $ x $ first and then $ y $. In this configuration, $ x $ ranges from $ y $ to 1, and $ y $ ranges from 0 to 1.

3. **Solving the Integral:**
- Choose one of the integral setups to evaluate and find the volume of the space under the surface described by $ f(x, y) $ over the mentioned triangular region.

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