Toolkit 103Applications of IntegralsArea and Volume by IntegralsA=∬R1 dAA=\iint_R 1\,dAA=∬R1dAV=∭D1 dVV=\iiint_D 1\,dVV=∭D1dVChoosing Limits of Integration∭f(x,y,z) dz dy dx\iiint f(x,y,z)\,dz\,dy\,dx∭f(x,y,z)dzdydxChoose the limits from inside to outside.z=z(x,y),y=y(x),x=constantz=z(x,y),\qquad y=y(x),\qquad x=\text{constant}z=z(x,y),y=y(x),x=constantExamplex=0→1,y=0→x2,z=0→yx=0\to1,\qquad y=0\to\frac{x}{2},\qquad z=0\to yx=0→1,y=0→2x,z=0→yV=∫01∫0x/2∫0ydz dy dxV=\int_0^1\int_0^{x/2}\int_0^y dz\,dy\,dxV=∫01∫0x/2∫0ydzdydx=∫01∫0x/2y dy dx=\int_0^1\int_0^{x/2}y\,dy\,dx=∫01∫0x/2ydydx=∫01x28 dx=124=\int_0^1\frac{x^2}{8}\,dx=\frac1{24}=∫018x2dx=241
