Unit 8: Applications of Integration
This unit turns integrals into geometry and accumulated quantities. You should be able to move flexibly between signed area, total change, volume, and other physical interpretations.
Area between curves
If \(f(x) \ge g(x)\) on \([a,b]\), then the area between the curves is
\[\int_a^b [f(x)-g(x)]\,dx.\]For horizontal slicing with respect to \(y\):
\[\int_c^d [x_{\text{right}}(y)-x_{\text{left}}(y)]\,dy.\][Image Placeholder: region between two curves with top-minus-bottom and right-minus-left labeling]
Accumulation and net change
If \(R(t)\) is a rate, then
\[\int_a^b R(t)\,dt\]gives net change over \([a,b]\).
So if \(P'(t)=R(t)\), then
\[P(b)-P(a)=\int_a^b R(t)\,dt.\]Distance vs displacement
If velocity is \(v(t)\), then:
\[\text{displacement} = \int_a^b v(t)\,dt\] \[\text{total distance} = \int_a^b \lvert v(t) \rvert\,dt\]Split total distance at sign changes of \(v(t)\).
Volume by cross sections
If cross-sectional area perpendicular to the axis is \(A(x)\), then volume is
\[V = \int_a^b A(x)\,dx.\]Common cross sections:
- squares,
- rectangles,
- semicircles,
- equilateral triangles.
Disk and washer methods
Disk method:
\[V = \pi \int_a^b [R(x)]^2\,dx\]Washer method:
\[V = \pi \int_a^b \left([R(x)]^2-[r(x)]^2\right)\,dx\][Image Placeholder: disk vs washer setup with labeled radii]
Volume by cylindrical shells
\[V = 2\pi \int_a^b (\text{radius})(\text{height})\,dx.\]Shell method is often simpler when washers would require solving for inverse functions or splitting many pieces.
Arc length
For a smooth function \(y=f(x)\) on \([a,b]\):
\[L = \int_a^b \sqrt{1+[f'(x)]^2}\,dx.\]This is useful BC-level enrichment even when not emphasized heavily in every AP class.
Improper integrals
An integral is improper if:
- an interval is infinite, or
- the integrand is unbounded.
Interpret through limits, for example:
\[\int_1^\infty \frac{1}{x^2}\,dx = \lim_{b \to \infty} \int_1^b \frac{1}{x^2}\,dx.\]Common mistakes
- Forgetting to split total distance when velocity changes sign.
- Using top-minus-bottom when the curves cross inside the interval without splitting.
- Using wrong radii in washer problems.
- Mixing shell and washer formulas without matching the slice geometry.