wk4 - MA104 Week 4 Applications of Integration Volume of a Solid of Revolution In general the volume of a solid is the sum of its cross-sectional areas

wk4 - MA104 Week 4 Applications of Integration Volume of a...

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MA104 Week 4 ° Applications of Integration Volume of a Solid of Revolution In general, the volume of a solid is the sum of its cross-sectional areas so that:V= limn!1nXi=1A(x°ix=RbaA(x)dx.Then, for example, if a solid region is generated by rotatingf(x)about thexaxis over the interval[a; b]the volume:V= limn!11Xi=1°[f(x)]2°x= limn!1b°an1Xi=1°°f±a+b°ani²³2=°Rba[f(x)]2dx:Using a similar thought process, the following is a summary of the methods used in determiningvolumes of revolutions:1. The disk method:(a) To determine the volume of a solid region generated by rotatingy=f(x)over the interval[a; b]about the:i)xaxis:ii)the liney=y0:V=°bZa[f(x)]2dxV=°bZa[f(x)°y0]2dx(b) To determine the volume of a solid region generated by rotatingx=g(y)over the interval[c; d]about the:i)yaxis:ii)the linex=x0:V=°dZc[g(y)]2dyV=°dZc[g(y)°x0]2dy2. The washer method:(a) To determine the volume of a solid generated by rotating the region betweenf(x)±g(x)±0over the interval[a; b]

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