1. Area Under a Curve
Riemann Sums

2. How do I find the area under a curve?b

3. think rectangles! We can easily find the area of a rectangle, right?
Can I divide the area under my curve into rectangles?

4. Start easy… Draw rectangles from the left endpoint.b
Draw rectangles from the left endpoint.
Draw rectangles from the right endpoint.
Draw rectangles from the middle.

5. Let f(x) = x2 + 1 from [0, 2], n = 4
Let’s approximate the area under the curve.
Graph it!
a = 0, b = 2 : [0, 2]
n = 4 (n = # of rectangles)
Divide your interval into 4 rectangles.

6. Let f(x) = x2 + 1 Draw rectangles from the left endpoint.
Compute area.
Area ~ A1 + A2 + A3 + A4
The is called the
Left Hand Sum (LHS)

7. Let f(x) = x2 + 1 Draw rectangles from the right endpoint.
Compute area.
This is called the Right Hand Sum (RHS)

8. Let f(x) = x2 + 1 Draw rectangles from the middle. Compute area.
This is called the Midpoint Sum

9. What did you find? Which area do you think is most exact?
Which is an under/over approximation?
Why?