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 !!!