Unit 8: Applications of Integration

Executive Summary

Unit 8 turns integration into geometry and physics: average values, net change, areas between curves, and volumes of solids.

Average value

$\frac{1}{b-a}\int_a^b f(x)\,dx$ — the constant height giving the same area. The Mean Value Theorem for Integrals guarantees $f(c)$ equals the average for continuous $f$.

Motion and accumulation

Displacement is $\int_a^b v(t)\,dt$; total distance is $\int_a^b |v(t)|\,dt$ (split where $v$ changes sign). Net change in any quantity equals the integral of its rate.

Area between curves

$\int_a^b (\text{top} - \text{bottom})\,dx$, with $a$ and $b$ from intersection points. For more than two intersections, split the interval and integrate each piece. If curves cross mid-interval, use $|f - g|$ or split.

Volumes

Disc method (rotate about an axis): $\pi \int (\text{radius})^2\,dx$. Washer method (two curves): $\pi \int (R^2 - r^2)\,dx$. Known cross sections: integrate the cross-section area $A(x)$ (squares, semicircles, equilateral triangles) perpendicular to the axis.

Exam traps

Average value divides by interval LENGTH, not area. Total distance requires $|v|$ — displacement alone can be zero for a round trip. Top-minus-bottom flips when curves cross. Washer radii are distances from the axis of rotation — account for axes other than the x-axis.