Applications of Integration
Core ideas
Integrals compute geometric and physical quantities. Area between curves is $\int (\text{top}-\text{bottom}),dx$ or $\int(\text{right}-\text{left}),dy$; choose the orientation that avoids splitting the region. Volumes come from slicing: disks/washers for rotation about an axis, $\pi\int(R^2-r^2),dx$; cylindrical shells when the axis is perpendicular to the slicing direction.
Arc length uses $\int\sqrt{1+(f')^2},dx$. Average value of a function and the Mean Value Theorem for integrals connect accumulation to typical value. Motion problems use integrals to recover displacement (integral of velocity) and total distance (integral of speed).
Traps
- Reversing top and bottom, or forgetting that curves can cross inside the interval (split the integral).
- Confusing washer inner and outer radii, especially when the axis of rotation is not a coordinate axis.
- Using the wrong variable of integration when curves are better described as functions of $y$.
- Displacement versus total distance: distance integrates $|v(t)|$.
Worked example
Find the area between $y=x$ and $y=x^2$ on $[0,1]$. The line is above the parabola there, so area $=\int_0^1(x-x^2),dx=\left[\frac{x^2}{2}-\frac{x^3}{3}\right]_0^1=\frac{1}{2}-\frac{1}{3}=\frac{1}{6}$.