Riemann Sums

Rectangles closing in on an area.

12 problems · about 22 minutes · LIM-5.A, LIM-5.B

What this lesson teaches

The student approximates area under a curve with Riemann sums and the trapezoidal rule.

Straightforward practice. Get the method working first.

5 problems

From 0 to 12 with 6 rectangles. Width of each?
Answer 2
Why 2.
What happens as the rectangles get thinner?
Answer The total approaches the exact area, which is the definite integral.
Why The sum converges on the exact area.
From 0 to 8 with 4 rectangles. Width of each?
Answer 2
Why 8 ÷ 4.
From 1 to 5 with 8 rectangles. Width of each, as a decimal?
Answer 0.5
Why 4 ÷ 8.
y = x² from 0 to 2, left sum with 2 rectangles. What is the estimate?
Answer 1
Why Heights 0 and 1, each 1 wide.

The same ideas with more to keep track of.

3 problems

y = x² from 0 to 2, right sum with 2 rectangles. What is the estimate?
Answer 5
Why Heights 1 and 4, each 1 wide.
On a decreasing function, does a right sum over or under estimate? 1 over, 2 under.
Answer 2
Why The right edge is the lowest point.
A velocity graph. What does the area beneath it represent? 1 distance, 2 acceleration.
Answer 1
Why Rate times time.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Sort each method by how it takes a rectangle height.
Answer From an edge: Left sum, Right sum · Not from a single edge: Midpoint sum, Trapezoidal rule
Why A trapezoid uses both edges, and a midpoint sum uses neither.
From 2 to 10 with 4 rectangles. Width of each?
Answer 2
Why 8 ÷ 4.
The Width: The interval from 0 to 6 is split into 3 rectangles. How wide is each?
Answer 2
Why 2.
The Direction: On an increasing function, does a left sum overestimate or underestimate? 1 over, 2 under.
Answer 2
Why It underestimates.