Calcule o volume do sólido obtido pela rotação, em torno do eixo x, do conjunto de todos os pares (x,y), tal que: 2x^2 + y^2 <= 1 e y >= 0.

O sólido formado pelas curvas acima é a metade um elipsóide de revolução em torno do eixo x, com limites em y de 0 a 1 e de x de -(x^(1/2))/2 e (x^(1/2))/2, portanto, o elemento de volume é:

dv=(pi/2).r^2 dx=(pi/2).y^2 dx, e o volume é:

v=(pi/2).Integral(y^2 dx)=(pi/2). Integral(1-2x^2) dx, os limites desta integral são -(x^(1/2))/2 e (x^(1/2))/2, então, após a integração o volume ficará:

V=(Pi/3) √2

\(\[\begin{align} & 2x{}^\text{2}\text{ }+\text{ }y{}^\text{2}\text{ }=\text{ }1 \\ & x\text{ }=\text{ }0\text{ }\Rightarrow \text{ }y\text{ }=\text{ }\pm \text{ }1 \\ & y\text{ }=\text{ }0\text{ }\Rightarrow \text{ }x{}^\text{2}\text{ }=\text{ }1/2\text{ }=\text{ }\pm \text{ }\surd \left( 1/2 \right)\text{ }\Rightarrow \text{ }x\text{ }=\text{ }\pm \text{ }\left( \surd 2 \right)/2 \\ & Temos: \\ & 2x{}^\text{2}\text{ }+\text{ }y{}^\text{2}\text{ }=\text{ }1 \\ & y{}^\text{2}\text{ }=\text{ }1\text{ }-\text{ }2x{}^\text{2} \\ & y\text{ }=\text{ }\pm \text{ }\surd \left( \text{ }1\text{ }-\text{ }2x{}^\text{2}\text{ } \right)\text{ }\to \text{ }y\text{ }\ge \text{ }0\text{ } \\ & y\text{ }=\text{ }\surd \left( \text{ }1\text{ }-\text{ }2x{}^\text{2}\text{ } \right)\text{ }=\text{ }\left( x \right) \\ & -\text{ }\frac{\left( \surd 2 \right)}{2}\le \text{ }x\text{ }\le \text{ }\frac{\left( \surd 2 \right)}{2} \\ & \surd 2|2 \\ & V\text{ }=\text{ }\pi \int \left[ \text{ }\left( x \right)\text{ } \right]{}^\text{2}\text{ }dx \\ & -\surd 2 \\ & 2 \\ & 0\text{ }\le \text{ }x\text{ }\le \text{ }\frac{\left( \surd 2 \right)}{2} \\ & \surd 2 \\ & 2V\text{ }=\text{ }2\pi \int \left[ \text{ }\left( x \right)\text{ } \right]{}^\text{2}\text{ }dx \\ & \surd 2 \\ & 2V\text{ }=\text{ }2\pi \int \left[ \surd \left( \text{ }1\text{ }-\text{ }2x{}^\text{2}\text{ } \right)\text{ } \right]{}^\text{2}\text{ }dx \\ & \surd 2 \\ & 2V\text{ }=\text{ }2\pi \int \left( \text{ }1\text{ }-\text{ }2x{}^\text{2}\text{ } \right)\text{ }dx \\ & \surd 2 \\ & 2V\text{ }=\text{ }2\pi .\left[ \text{ }x\text{ }-\text{ }\left( \frac{2x{}^\text{3}}{3} \right)\text{ } \right] \\ & \\ \end{align}\] \)
\(\[\begin{align} & V\text{ }=\text{ }2\pi .\left\{ \frac{\text{ }\left[ \text{ }\frac{\left( \surd 2 \right)}{2}\text{ } \right]\text{ }-\text{ }2\left[ \text{ }\frac{\left( \surd 2 \right)}{2}\text{ } \right]{}^\text{3}}{3}\text{ } \right\} \\ & V\text{ }=\text{ }2\pi .\left\{ \text{ }\frac{\left[ \text{ }\frac{\left( \surd 2 \right)}{2}\text{ } \right]\text{ }-\text{ }\left[ \frac{\text{ }\left( 2\surd 8 \right)}{8}\text{ } \right]}{3} \right\} \\ & V\text{ }=\text{ }2\pi .\left\{ \text{ }\left[ \text{ }\frac{\left( \surd 2 \right)}{2}\text{ } \right]\text{ }-\text{ }\left[ \text{ }\frac{\left( 2\surd 8 \right)}{24}\text{ } \right]\text{ } \right\} \\ & V\text{ }=\text{ }2\pi .\left\{ \text{ }\left[ \text{ }\frac{\left( \surd 2 \right)}{2}\text{ } \right]\text{ }-\text{ }\left[ \text{ }\frac{\left( 2.2\surd 2 \right)}{24}\text{ } \right]\text{ } \right\} \\ & V\text{ }=\text{ }2\pi .\left\{ \text{ }\left[ \text{ }\frac{\left( \surd 2 \right)}{2} \right]\text{ }-\text{ }\left[ \text{ }\frac{\left( 4\surd 2 \right)}{24}\text{ } \right]\text{ } \right\} \\ & V\text{ }=\text{ }2\pi .\left\{ \text{ }\left[ \text{ }\frac{\left( \surd 2 \right)}{2}\text{ } \right]\text{ }-\text{ }\left[ \text{ }\frac{\left( \surd 2 \right)}{6}\text{ } \right]\text{ } \right\} \\ & V\text{ }=\text{ }2\pi .\left\{ \text{ }\frac{\left[ \text{ }3.\left( \surd 2 \right)\text{ }-\text{ }\left( \surd 2 \right)\text{ } \right]}{6}\text{ } \right\} \\ & V\text{ }=\text{ }2\pi .\left\{ \text{ }\frac{\left[ \text{ }2.\left( \surd 2 \right)\text{ } \right]}{6}\text{ } \right\} \\ & R\text{ }\to V\text{ }=\text{ }\frac{\left( \text{ }2\pi .\surd 2\text{ } \right)}{3}u.v \\ \end{align}\] \)