Riemann Sums

Rectangles closing in on an area.

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

NameDate

Figure — use these to answer the problems

Move the two points and watch the shaded region. That area is what a Riemann sum estimates.

y = 1x² + 0x + 0

Left edge0

Right edge3

Figure 1

Straightforward practice. Get the method working first.

5 problems

From 0 to 12 with 6 rectangles. Width of each?
Answer
2
What happens as the rectangles get thinner?
- The total approaches the exact area, which is the definite integral.
- The estimate becomes less reliable.
From 0 to 8 with 4 rectangles. Width of each?
Answer
From 1 to 5 with 8 rectangles. Width of each, as a decimal?
Answer
y = x² from 0 to 2, left sum with 2 rectangles. What is the estimate?
Answer

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
On a decreasing function, does a right sum over or under estimate? 1 over, 2 under.
Answer
A velocity graph. What does the area beneath it represent? 1 distance, 2 acceleration.
Answer

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.
Write each item under the heading it belongs to: Left sum · Right sum · Midpoint sum · Trapezoidal rule
From an edge
Not from a single edge
From 2 to 10 with 4 rectangles. Width of each?
Answer
The Width
The interval from 0 to 6 is split into 3 rectangles. How wide is each?
Answer
The Direction
On an increasing function, does a left sum overestimate or underestimate? 1 over, 2 under.
Answer