Use Fubini’s Theorem for an \(y\)-Simple Region

Find \(\iint\limits_{\kern-3ptR}3x^{2}y\,dA\) if \(R\) is the region bounded by the smooth curves \(x=\sqrt{y}\) and \(x=-y\), and the line \(y=1\).

Solution We begin by graphing the region \(R\). See Figure 16. Observe that \(R\) is a closed, bounded region that is \(y\)-simple, where \( h_{1}(y)=-y\) and \(h_{2}(y)=\sqrt{y}\), \(0\leq y\leq 1\). Using Fubini’s Theorem for a \(y\)-simple region, we have \begin{eqnarray*} \displaystyle\iint\limits_{\kern-3ptR}3x^{2}y\,dA &=&\int_{0}^{1}\left[ \int_{-y}^{\sqrt{y}}3x^{2}y\,dx\right] \! dy=\int_{0}^{1}\big[ y\,x^{3}\big] _{-y}^{\sqrt{y}}\,dy\\[4pt] &=&\int_{0}^{1}y(y^{3/2}+y^{3})\,dy=\int_{0}^{1}(y^{5/2}+y^{4})\,dy \\[4pt] &=&\left[ \dfrac{y^{7/2}}{7/2}+\dfrac{y^{5}}{5}\right] _{0}^{1}=\dfrac{2}{7}+\dfrac{1}{5}=\dfrac{17}{35} \end{eqnarray*}