1. 5.1 Calculus and Area

2. I. Problem!
Given a nonnegative function f (x), i.e., f (x) > 0, on [a, b]. Find the area bounded by the curve, the x- axis, and the vertical lines x = a and x = b.

3. II. Estimation of the Area
A.) Partition [a, b] into n subintervals of equal length by choosing x-values

4. B.) Evaluate the area for given n using rectangles:
Rectangular Approximation Method (RAM)
Left-hand approximation:

5. Right-hand approximation:
Mid-point approximation:

6. C.) Visually: 1 2 3 4
LRAM

7. 1 2 3 4
RRAM

8. 1 2 3 4
MRAM

9. III. Solution – Area Under a Curve
Def.- If f (x) ≥ 0 and is continuous on [a, b], then the area bounded by the curve, the x-axis, and the vertical lines x = a and x = b is

10. IV. Examples of Approximations of Area
A.) Approximate the area under the following curve from x = 1 to x = 3 using 4 rectangles and all three RAMs.

11. LRAM

12. RRAM

13. MRAM

14. B.) Approximate the area under the following curve from x = 0 to x = 2 using 10 rectangles and all three RAMs.

15. Is LRAM always an underestimate(lowest)?
C.) Approximate the area under the following curve from x = 0 to x = 5 using 3 rectangles and all three RAMs.