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Question Number 189374 by mathocean1 last updated on 15/Mar/23

$$\mathrm{\Delta }=\{(x,y,z)\in {\mathbb{R}}^3:x^2+y^2⩽1,x⩾0,0⩽z⩽1+y\}.$$$$calculate:$$$$\int \int {\int }_{\mathrm{\Delta }}dxdydz.$$

Answered by Ar Brandon last updated on 15/Mar/23

$$={\int }_0^1{\int }_0^{\sqrt{1-x^2}}{\int }_0^{1+\mathrm{y}}dzd\mathrm{y}dx={\int }_0^1{\int }_0^{\sqrt{1-x^2}}(1+\mathrm{y})d\mathrm{y}dx$$$$={\int }_0^1{[\mathrm{y}+\frac{{\mathrm{y}}^2}{2}]}_0^{\sqrt{1-x^2}}dx={\int }_0^1(\sqrt{1-x^2}+\frac{1-x^2}{2})dx$$

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