The Fundamental Theorem of Calculus Area and The Definite Integral OBJECTIVES  Evaluate a definite integral.  Find the area under a curve over a given closed interval.  Interpret an area below the horizontal axis.  Solve applied problems involving definite integrals.

Fundamental theorem of calculus: for a continuous function f on the interval[a, b] where F is any anti-derivative of f. a and b are called the lower and upper limits of integration.

The notation F(b) - F(a) means to evaluate the anti-derivative at b and subtract the anti-derivative evaluated at a. Since both F(b) and F(a) contain the constant of integration c, they will cancel each other out, thus eliminating c altogether.

Example: Evaluate each of the following:

Example (continued):

Evaluate each definite integral

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Properties of the Definite Integral (c is a constant)

The Definite Integral Using the Properties of the Definite Integral Given:

The Fundamental Theorem of Calculus, Part 2 If f is continuous on then the function has a derivative at every point in, and

Fundamental Theorem: 1. Derivative of an integral.

2. Derivative matches upper limit of integration. First Fundamental Theorem: 1. Derivative of an integral.

2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant. First Fundamental Theorem:

1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant. New variable. Fundamental Theorem:

1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant. The long way: Fundamental Theorem:

1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant.

The upper limit of integration does not match the derivative, but we could use the chain rule.

The lower limit of integration is not a constant, but the upper limit is. We can change the sign of the integral and reverse the limits.

Neither limit of integration is a constant. It does not matter what constant we use! (Limits are reversed.) (Chain rule is used.) We split the integral into two parts.

AREA UNDER A CURVE

When we find the area under a curve by adding rectangles, the answer is called a Rieman sum. subinterval partition The width of a rectangle is called a subinterval. The entire interval is called the partition. Subintervals do not all have to be the same size.

subinterval partition If the partition is denoted by P, then the length of the longest subinterval is called the norm of P and is denoted by. As gets smaller, the approximation for the area gets better. if P is a partition of the interval

is called the definite integral of over. If we use subintervals of equal length, then the length of a subinterval is: The definite integral is then given by:

Leibnitz introduced a simpler notation for the definite integral: Note that the very small change in x becomes dx.

Area

IF then the definite integral denoted by is defined to be the area of the region between the curve of f(x) and the x-axis bounded by the vertical lines at a and b

Area from x=0 to x=1 Example: Find the area under the curve from x = 1 to x = 2. Area from x=0 to x=2 Area under the curve from x = 1 to x = 2.

Determine the area under the given curve for the values of x.

Geometric Interpretation Area of R 1 – Area of R 2 + Area of R 3 a b R1R1 R2R2 R3R3 BUT WHAT HAPPENS IF THE GRAPH DIPS BELOW THE X-AXIS?? AS AREA CANNOT BE NEGATIVE!!!

When finding area below the x-axis from [a, b], if you just find the definite integral, you will get a negative answer. Area can’t be negative, so just take the absolute value of the definite integral or put a – in front of it. If f(x) is negative for the values of x in the interval [a, b], take the opposite or the absolute value of the definite integral to get the area.

Example: Find the area between the x-axis and the curve from to. pos. neg. 

To find the area enclosed by the curve and the x-axis, we must separate the interval by the x-intercepts and integrate accordingly

\ Using Geometry to Compute the Integral Ex. Use geometry to compute the integral Area = 2 Area =4

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Area Between Two Curves a b Let f and g be continuous functions, the area bounded above by y = f (x) and below by y = g(x) on [a, b] is provided that R

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Ex. Find the area bounded by the curves and the vertical lines x = – 1 and x = 2.

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Average Value of a Function If f is integrable on [a, b], then the average value of f over [a, b] is Ex. Find the average value of