Cogito
Calculus · Chapter 7 · Lesson 1
Area Under a Curve
Accumulation, before any antiderivative.
10 problems · about 22 minutes · LIM-5.A, LIM-5.B
What this lesson teaches
The student interprets the definite integral as accumulated area and estimates it with Riemann sums.
- Area under a rate graph is an accumulated total.
- Rectangles estimate it; thinner ones estimate it better.
- The definite integral is the limit of those sums.
Warm-Up Review
From earlier lessons. Loosen up before the new work.
2 problemsReview — Optimisation: Two numbers add to 16. What is their greatest product?
Answer 64
Why 64.
Review — The Second Derivative and Concavity: f′(4) = 0 and f″(4) = −3. Max or min? 1 max, 2 min.
Answer 1
Why A maximum.
Warm Up
Straightforward practice. Get the method working first.
4 problemsSpeed rises 0 to 12 m/s over 10 s. Distance?
Answer 60
Why 60 m.
What is a definite integral?
Answer The limit of sums of ever-thinner rectangles.
Why A limit of Riemann sums.
A car at 12 m/s for 5 s. Distance?
Answer 60
Why 12 × 5.
Speed rises 0 to 8 m/s over 4 s. Distance?
Answer 16
Why Half of 4 × 8.
Build It Up
The same ideas with more to keep track of.
2 problemsA car at 25 m/s for 8 s. Distance?
Answer 200
Why 25 × 8.
Under a speed-time graph, area represents what?
Answer Distance.
Why Speed times time.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
2 problemsThe Steady Speed: A car at 15 m/s for 4 seconds. What distance?
Answer 60 m
Why 60 m.
The Triangle: Speed rises steadily from 0 to 10 m/s over 6 seconds. What distance?
Answer 30 m
Why 30 m.