1. 6-4 Day 1 Fundamental Theorem of Calculus

2. The Fundamental Theorem of Calculus has 2 parts
*The Fundamental Theorem of Calculus has 2 parts. One part relates derivatives and integrals and the other establishes how we evaluate integrals by hand.
Fundamental Theorem of Calculus: Part 1
Every continuous function f is deriv of some other function
Every continuous function has an antiderivative
Integration & differentiation are inverses of each other

3. Ex 1) Using the Fundamental Theorem of Calculus, find:
a) b)
Ex 2) Find
(Use chain rule!)

4. Ex 3) Find a) b)
(variable lower limits)

5. Ex 4) Find a function y = f (x) with derivative
that satisfies the condition f (3) = 5.
Since y (3) = 0
We need to add 5
*Note: We can graph the integral of a function by using fnInt Just put x as the upper limit.

6. Fundamental Theorem of Calculus: Part 2
(Integral Evaluation Theorem)
*says any definite integral of any continuous function can be calculated without taking limits or Riemann sums – so long as antiderivative can be found
Ex 5) Evaluate
(can check with fnInt)

7. Ex 6) Find the area of the region between the curve
y = 4 – x2 , 0  x  3, and the x-axis.

8. To find Total Area Analytically
Partition using zeros
Integrate each subinterval
Add up absolute values
To Find Total Area Numerically
Use fnInt with absolute value
Ex 7) Find the area of the region between the curve
y = x cos 2x and the x-axis over the interval –3  x  3
Use Math  9: fnInt (abs (x cos 2x), x, –3, 3)
abs value Math  Num  1

9. homework
Pg. 294 # 2, 6, 11, 18, 21, 23, 27, 31, 32, 49
Pg. 306 # 1–49 (mult of 1+3n)