Considere o sólido definido pelo conjunto S=\{(x,y,z)\in\mathbb{R}^3:x\geq 0,y\geq 0,z \leq 4, y\leq \sqrt{3-x^2},z\geq x\} .

Question

Considere o sólido definido pelo conjunto  S=\{(x,y,z)\in\mathbb{R}^3:x\geq 0,y\geq 0,z \leq 4, y\leq \sqrt{3-x^2},z\geq x\} .

Answer

S=\displaystyle \left\{(r,	heta,z)\in \mathbb{R}^3:0 \leq 	heta \leq \frac{\pi}{2}, 0\leq r \leq \sqrt{3}, r\cos(	heta)\leq z \leq 4 ight\} .

Answer

\displaystyle \iiint_S xz \ dV= \displaystyle \int_0^{\sqrt{3}}\int_0^{\frac{\pi}{2}} \int_{r \cos(	heta)}^4 zr^2\cos(	heta) dzd	heta dr .

Answer

Vol(S)= \displaystyle \iint_D \left(\int_x^4 1 \ dz \right) dxdy  onde   D=\left\{ (x,y) \in \mathbb{R}^2: x^2+y^2\leq 3, x\geq 0, y \geq 0 \right\}.

Answer

Vol(S)=\displaystyle \int_0^{\frac{\pi}{2}} \int_0^{\sqrt{3}} r^2\cos(	heta) drd	heta .

Answer

Vol(S)=\displaystyle \frac{2\sqrt{3}\pi}{3}.

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