Unit 8: Applications of Integration
1. Average value of a function (free preview)
The average value of $f$ on $[a, b]$ spreads its total accumulation evenly over the interval:
$f_{avg} = \dfrac{1}{b - a}\displaystyle\int_a^b f(x)\,dx$
Picture it as water finding its level: the area under the curve, reshaped into a rectangle of the same width. This formula is one of the exam's most reliable easy points, and one of its most reliably fumbled.
Worked example
Find the average value of $f(x) = \sin x$ on $[0, \pi]$.
- Total: $\displaystyle\int_0^\pi \sin x\,dx = 2$.
- Divide by the length: $\dfrac{2}{\pi}$.
- Sanity check: $\frac{2}{\pi} \approx 0.64$, between the min 0 and max 1. Reasonable.
Common mistake: forgetting the $\frac{1}{b-a}$ and reporting the whole integral, or averaging the two endpoint values. The integral is the total; the average divides by the width.
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The rest of Unit 8 is in the full course
The final unit, home of the famous volume problems:
- Area between curves, including sideways (dy) regions and crossings
- Disk and washer volumes, including off-axis lines like $y = -1$
- Known cross sections: squares, rectangles, triangles (and when $\pi$ does NOT belong)
- Motion: position from velocity, displacement vs total distance
- 45 exam-style questions with verified step-by-step solutions
All 8 units, 355 questions, 2 full-length exams. 7-day refund.