Applications of Integration
What you must understand
Use definite integrals to compute average value, $\frac{1}{b-a}\int_a^b f(x),dx$, and to find areas between curves: $\int_a^b (\text{top}-\text{bottom}),dx$ or right-minus-left with respect to $y$.
Volumes: disks/washers use $\pi\int [R(x)^2-r(x)^2],dx$ for rotation about an axis; cross-section methods integrate area $A(x)$. Set up the integral from a sketch, identifying outer and inner radii or the correct bounds.
Accumulation and net change: integrating a rate gives total change; total distance uses $\int|v(t)|,dt$. Motion problems connect position, velocity, and displacement through integration.
Traps
- Subtracting curves in the wrong order, producing negative area.
- Forgetting $\pi$ or squaring radii in disk/washer setups.
- Using the wrong bounds when curves cross (split the integral).
- Confusing displacement with total distance.
- Ignoring the axis of rotation when determining radii.
Worked example
Area between $y=x$ and $y=x^2$ on $[0,1]$: top is $x$, so $\int_0^1(x-x^2),dx=[\frac{x^2}{2}-\frac{x^3}{3}]_0^1=\frac12-\frac13=\frac16$.
Rule of thumb: sketch first, decide top-minus-bottom (or outer/inner radii), and split at intersection points; check that areas and volumes are positive.