Section 6.2: Volumes
- Slice solid into cross-sections
- Approximate slices with “cylinders”
$v(k^{th} slice) \approx V(k^{th} cylinder)$
$= A(X_k) * \Delta X_k$
$V \approx \sum V_k$
$= \sum A (X_k) * \Delta X_k$ Reumann Summ
Take Limit -> $\int_a^b A(x) dx$
Volume of solid with cross-sectional area $A(X)$ is $V=\int_a^b A(X) dx$
Solid of Revolution
- rotate plan region aorund axis
- slices are disks
$A(x) = \pi r^2 = \pi[R(x)]^2$
$V = \pi \int_a^b[R(x)]^2 dx$
Example 1:
$R(x) = e^x$, $[0, 2]$, about x-axis
$V=\pi \int_0^2 (e^x)^2$
NOT DONE