The Definite Integral and The Fundamental Theorem of Calculus

This method used the sum of the area of intervals under a curve- called Reimann Sums

The limit of the sums of intervals is the same as a definite integral over the same interval.

b

A (x)

• A’ (x) = f (x)• A (a) = 0 and F (x) = A (x) + C• A (b) = A

The Fundamental Theorem of Calculus, Part I

How about some practice?

More Examples !!!

Evaluate If

TOTAL AREA

A1 A3 A5

a A2 A4 b

Practice Time !!!Find the total area between the curve y = 1 – x2 and the x-axis over the interval [0, 2].

The Mean Value Theorem for Integrals:

Over any interval, there exists an x value which creates a y value that is the height of a rectangle which will equal the area under the curve.

The Average Value:

The function value, f(c), found by the Mean Value Theorem

Example

In analyzing the graph of F(x) we would look at the derivative:

f (x)

The Fundamental Theorem of Calculus, Part II

How about some practice?

Integrals with Functions as Limits of Integration

Let’s Practice !!!