Question
Download Solution PDFThe value of \(\displaystyle\int_0^1 \int_0^x (x^2 + y^2)dA\), where dA indicate small area in xy-plane, is
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
\(\displaystyle\int_0^m \int_0^x f(x,y)dA\)
Since the inside limit is in terms of x, therefore we have to integrate first the 'y' terms and convert whole expression in terms of x.
Calculation:
Given:
\(\displaystyle\int_0^1 \int_0^x (x^2 + y^2)dA\)
\(\displaystyle\int_0^1\left[ \int_0^x (x^2 + y^2)dy\right]dx\;\)
\(\displaystyle\int_0^1\left( x^2[y]_0^x\;+\;\left[\frac{y^3}{3}\right]_0^x\right)dx\;\)
\(\displaystyle\int_0^1\left( x^3\;+\;\frac{x^3}{3}\right)dx\;\)
\(\displaystyle\int_0^1\left(\frac{4x^3}{3}\right)dx\;\)
\(\displaystyle \left(\frac{4x^4}{12}\right)_0^1\;\)
\(\displaystyle \frac{1}{3}\)
Last updated on May 28, 2025
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