6.5 & 6.6 & 6.9 the definite integral and the fundemental theorem of calculus copy
TRANSCRIPT
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 !!!