Compute the double integral of f(x, y) = 6x²y over the given shaded domain in the following Figure : 2 1 y 3 4 X S6x²y dA= 236.8
Compute the double integral of f(x, y) = 6x²y over the given shaded domain in the following Figure : 2 1 y 3 4 X S6x²y dA= 236.8
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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![**Transcription for Educational Website**
**Topic: Double Integrals**
---
**Problem Statement:**
Compute the double integral of \( f(x, y) = 6x^2y \) over the given shaded domain in the following figure.
**Illustration:**
*Graph Description:*
A graph is displayed with the x-axis ranging from 0 to 4 and the y-axis from 0 to 2. The shaded region, which is the domain of integration, is a right triangle. The base of the triangle lies on the x-axis from \( x = 1 \) to \( x = 4 \). The height of the triangle extends vertically to \( y = 2 \) at \( x = 4 \), creating the hypotenuse from \( (1, 0) \) to \( (4, 2) \).
---
**Solution:**
\[
\iint_D 6x^2 y \, dA = 236.8
\]
---
This covers the computation of the double integral over a specified domain using the function \( f(x, y) = 6x^2y \), where the region of interest is depicted as a triangle on a Cartesian plane. The resultant value of the double integral is 236.8.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd3444a3f-7d5c-4474-ab08-7930ed9e066d%2F5c5a3ed6-e37b-4add-ab88-c5994431ae89%2Ftz72grd_processed.png&w=3840&q=75)
Transcribed Image Text:**Transcription for Educational Website**
**Topic: Double Integrals**
---
**Problem Statement:**
Compute the double integral of \( f(x, y) = 6x^2y \) over the given shaded domain in the following figure.
**Illustration:**
*Graph Description:*
A graph is displayed with the x-axis ranging from 0 to 4 and the y-axis from 0 to 2. The shaded region, which is the domain of integration, is a right triangle. The base of the triangle lies on the x-axis from \( x = 1 \) to \( x = 4 \). The height of the triangle extends vertically to \( y = 2 \) at \( x = 4 \), creating the hypotenuse from \( (1, 0) \) to \( (4, 2) \).
---
**Solution:**
\[
\iint_D 6x^2 y \, dA = 236.8
\]
---
This covers the computation of the double integral over a specified domain using the function \( f(x, y) = 6x^2y \), where the region of interest is depicted as a triangle on a Cartesian plane. The resultant value of the double integral is 236.8.
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