MATH 152 Chapter 7.2

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Volume by Slicing

Volume of a hemisphere: ${\frac {2}{3}}\pi r^{3}$

Disks

To find a volume of a solid generated by revolving a region around the x-axis:

$\pi \int _{a}^{b}{r^{2}\,\mathrm {d} x}$

Washers

If the region has a "hole" when revolved, be sure to subtract the "inner" volume from the "outer" volume:

$\pi \int _{a}^{b}{{r_{\mathrm {out} }}^{2}-{r_{\mathrm {in} }}^{2}\,\mathrm {d} x}$

LEARN IDENTITIES IN APPENDIX D

Slices of a given shape

region:

• $y=x^{2}$
• $y=2-x^{2}$

Cross-sections are squares ($A=s^{2}$), and slices will be very thin rectangular prisms. Therefore:

• $V=lwh$
  
• $h=\mathrm {d} x$
  
• $l=w=(2-x^{2})-x^{2}$

$V=\int _{-1}^{1}{((2-x^{2})-x^{2})^{2}\,\mathrm {d} x}=\int _{-1}^{1}{(2-2x^{2})^{2}\,\mathrm {d} x}$

By symmetry:
$V=2\int _{0}^{1}{4-8x^{2}+4x^{4}\,\mathrm {d} x}$