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4.4 The Fundamental Theorem of Calculus
Evaluating a Definite Integral
• Evaluate each definite integral.
2 4 4
2 2
1 1 0
x 3 dx 3 xdx sec xdx
A definite integral involving absolute value
2
0
2x 1 dx
Using the fundamental theorem to find area
• Find the area of the region bounded by the graph , the x-axis, and the vertical lines x=0 and x=2.
2y 2x 3x 2
Using Mean Value Theorem
• Find the values of c guaranteed by the mean value theorem for integrals over the given interval. 3
9f (x) , 1,3
x
Finding the Average Value of a Function
• Find the average value of on the interval [1,4].
2f (x) 3x 2x
The Definite Integral as a Function
• Evaluate the function
at
x
0
F(x) cos tdt
x 0, , , , and6 4 3 2
Using the Second Fundamental Theorem of Calculus
x2
0
dt 1dt
dx
Using the Second Fundamental Theorem of Calculus
• Find the derivative of 3x
2
F(x) cos tdt