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