Volumes from Areas of Known Cross Sections (College Board AP® Calculus AB): Revision Note

Volumes from areas of known cross sections

How do I find the volume of a solid with known cross sections?

• If the area of the cross section of a solid

  • can be expressed as a function of $\mathit{x}$

    • i.e. as a function $A$ that may be written in the form $A\left(x\right)$

  • and if $A\left(x\right)$ is continuous on the interval $\left[a,b\right]$

• Then the volume of the corresponding solid from $x=a$ to $x=b$ is

  • $\mathrm{Volume}={\int }_a^{b}A\left(x\right)dx$

• For example the cross-sectional area of a solid is given by $A\left(x\right)=\frac{1}{x+1},0\le x\le 3$

  • The volume of the solid between $x=0$ and $x=3$ is

    • $\mathrm{Volume}={\int }_0^3\frac{1}{x+1}dx={\left[\ln \left(x+1\right)\right]}_0^3=\ln 4-\ln 1=\ln 4-0=\ln 4$

• This method of finding volumes uses the idea of a definite integral as calculating an accumulation of change

  • $A\left(x\right)·∆x$ is the volume of a solid with cross-sectional area $A\left(x\right)$ and length $∆x$

  • $A\left(x\right)dx$ is the limit of this volume element as $∆x\to 0$

  • The integral ${\int }_a^{b}A\left(x\right)dx$ sums up all these infinitesimal volume elements between $x=a$ and $x=b$