1. Chapter 5
AP Calculus BC

2. 5.1 Estimating with Finite Sums
Drive 2 hours at 75 mph…..Distance Traveled?? Graph….
Graph it for x from 0 to 5
5 subintervals……..
Example:
LRAM
RRAM
MRAM
Do some examples:

3. 5.2 Definite Integrals Riemann Sums convert to Integrals….
Parts of an Integral:
a is the lower limit
b is the upper limit
f(x) is the Integrand
dx – variable of integration
All continuous functions are integrable.
Area under a curve – If y = f(x) is nonnegative and integrable over a closed interval [a,b] then the area under the curve y = f(x) from a to b is:

4. 5.2 cont’d.
When f(x)<0, then the integral will come out negative.
Area:
Area above – area below
If f(x) = c, where c is a constant, on the interval [a,b] then:
Calculator – fnint(f(x), x,a,b)
Examples:
Riemann Sums

5. 5.3 Definite Integrals/Antiderivatives
Properties:
Order of Integration:
Zero:
Constant Multiple:
Sum/Difference:
Additivity:
Find:
Example: Given:

6. 5.3 cont’d.
Average Value Theorem- If f is integrable on [a,b] its average value on [a,b] is
Find av(f) and does f ever take on this value in the interval?
Mean Value Theorem for Definite Integrals – If f is cts. On [a,b] then at some point c in [a,b]
Fundamental Theorem of Calculus
More in 5.4

7. 5.4 Fundamental Theorem of Calculus
EXAMPLES:
Part 2: Integral Evaluation Theorem
Evaluate:
Find the area of the region between
And the x-axis on [0,3]

8. 5.5 Trapezoidal Rule WHY? To Approximate: Use the Trapezoidal Rule…
Examples:
Use Trap. Rule with n=4………
Find the actual answers.