Download 5.4 Fundamental Theorem of Calculus Herbst - Spring

5-4 Fundamental Theorem of
Calculus
Part 2
• If f is continuous at every point of [a, b] and if
F is any antiderivative of f on [a, b], then
Example
• Set up the equation to find the area of the
4
shaded region.
F ( x )   (  x  4 x  4)dx
2
x3
Part 1
• If f is continuous on [a, b], then the function
has a derivative at every point x in [a, b], and
This means… you actually
do nothing… no derivative,
not antiderivative…. Just
switch variables!
Example
• Find
But what if the bound has a derivative???
Part 1
• If f is continuous on [a, b], then the function
has a derivative at ever point x in [a, b], and
Example
• Find
Let u = x2
Example
• Find dy/dx
Must have the
variable as the upper
bound so….
• Find dy/dx
Need to split so only one
variable in each and rewrite
so both have the variable as
upper bound
Example
• Find dy/dx
Example
• Find dy/dx
Constructing a Function
• Find a function y such that dy/dx = tan x and
satisfies the condition f(3) = 5
Definite Integrals on the Calculator
• Evaluate
Homework
• P. 302 10-16, 45, 48, 52