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$\int_0^a \int_0^{\sqrt{a^2 -x^2}} x^2 y\,dx\,dy$

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Simplify radical expressions involving fractions

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Q.

\int_0^a \int_0^{\sqrt{a^2 -x^2}} x^2 y\,dx\,dy

Identify Limits and Function: Identify the limits of integration and the function to integrate. $\newline$ We are integrating $x^2 y$ with respect to $y$ from $0$ to $\sqrt{a^2 - x^2}$ and with respect to $x$ from $0$ to $a$ .
Integrate with respect to $y$ : Integrate with respect to $y$ first. $\int_{0}^{\sqrt{a^2 - x^2}} x^2 y \, dy = x^2 \left[\frac{1}{2} y^2\right]_{0}^{\sqrt{a^2 - x^2}}$ = $x^2 \cdot \frac{1}{2} \cdot (a^2 - x^2)$ = $\frac{1}{2} x^2 (a^2 - x^2)$
Integrate with respect to $x$ : Now integrate the result with respect to $x$ . $\int_{0}^{a} \frac{1}{2} x^2 (a^2 - x^2) \,dx$ This requires expanding and then integrating term by term. $= \frac{1}{2} \int_{0}^{a} (a^2x^2 - x^4) \,dx$ $= \frac{1}{2} \left[\frac{a^2}{3} x^3 - \frac{1}{5} x^5\right]_{0}^{a}$ $= \frac{1}{2} \left[\frac{a^5}{3} - \frac{a^5}{5}\right]$ $= \frac{1}{2} \left[\frac{2a^5}{15}\right]$ $= \frac{a^5}{15}$

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