Why this chapter matters
These applications show how integration answers total-amount questions that derivatives alone cannot.
What you will learn
- Compute net and total change from rate functions on intervals.
- Find area between curves using intersection bounds.
- Model accumulation in position, revenue, and other applied settings.
Lessons in this chapter
- Accumulation from ratesConvert a known rate function into total change over time.
- Area between two curvesSet up top-minus-bottom integrals with correct bounds.
- Volume from cross sectionsIntegrate known cross-sectional area formulas across an interval.
- Average value of a functionUse integral mean value to summarize behavior on an interval.
Study task
Two functions model cost and revenue rates. Integrate each over the same interval, compare totals, and interpret the difference in context.
Chapter checkpoint
If v(t) = 6t - t^2 on 0 <= t <= 4, what is the displacement on [0, 4]?
Displacement is integral from 0 to 4 of (6t - t^2) dt = [3t^2 - t^3/3] from 0 to 4 = 48 - 64/3 = 80/3.