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\(\int\limits^1_0\int\limits^x_0(x^2 + y^2 )\,dx\,dy\), Find its value? |
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Answer» \(\int\limits^1_0\int\limits^x_0(x^2 + y^2 )\,dx\,dy\) \(= \int\limits^1_0\left[\int\limits^x_0x^2+ y^2 \,dy\right]dx\) \(= \int\limits^1_0\left[x^2y+\frac{y^3}{3}\right]^x_0\, dx\) \(= \int\limits^1_0\left(x^3 + \frac{x^3}{3}\right)dx\) \(= \frac43\int\limits^1_0x^3\,dx\) \(= \frac43 \left[\frac{x^4}{4}\right]^1_0\) \(= \frac43\times\frac14\) \(= \frac13\) |
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