1. Section 4.3 – Riemann Sums and Definite Integrals

2. Riemann Sums
The rectangles need not have equal width, and the height may be any value of f(x) within the subinterval.
1. Partition (divide) [a,b] into N subintervals.
2. Find the length of each interval:
3. Find any point ci in the interval [xi,xi-1]. c1 c2 c3 ci cN
=x0 a x1 x2 x3 xi
=xN b
4. Construct every rectangle of height f(ci) and base Δxi.
4. Find the sum of the areas.

3. Riemann Sums
The norm of P, denoted ││P││, is the maximum of the lengths Δxi. a b
As ││P││ gets closer to 0, the sum of the areas of the rectangles is closer to the actual area under the curve

4. Riemann Sums
The norm of P, denoted ││P││, is the maximum of the lengths Δxi. a b
As ││P││ gets closer to 0, the sum of the areas of the rectangles is closer to the actual area under the curve

5. Upper limit of integration Lower limit of integration
Definite Integral
The definite integral of f(x) over [a,b], denoted by the integral sign, is the limit of Riemann sums: Where the limit exists, we say that f(x) is integrable over [a,b].
Upper limit of integration
Lower limit of integration

6. Notation Examples
The definite integral that represents the area is… EX1:
f(x) S a b
Ex2: The area under the parabola y=x2 from 0 to 1

7. Theorem: The Existence of Definite Integrals
If f(x) is continuous on [a,b], or if f(x) is continuous with at most finitely many jump discontinuities (one sided limits are finite but not equal), then f(x) is integrable over [a,b]. a b

8. Negative Area or “Signed” Area
If a function is less than zero for an interval, the region between the graph and the x-axis represents negative area.
Positive Area
Negative Area

9. Definite Integral: Area Under a Curve
If y=f(x) is integrable over a closed interval [a,b], then the area under the curve y=f(x) from a to b is the integral of f from a to b.
Upper limit of integration
Lower limit of integration

10. Example 1
Calculate

11. Example 2
Calculate

12. Rules for Definite Integrals
Let f and g be functions and x a variable; a, b, c, and k be constant.
Constant
Constant Multiple
Sum Rule
Reversing the Limits
Additivity

13. Constant Multiple Rule Given and Constant Rule
Example 1
If , calculate .
Sum Rule
Constant Multiple Rule
Given and Constant Rule

14. Example 2
If , calculate
Additivity Rule
Given