6.5 The Definite Integral In our definition of net signed area, we assumed that for each positive number n, the Interval [a, b] was subdivided into n subintervals of equal length to create bases for the approximating rectangles. For some functions, it may be more convienct to use rectangles with different width. Then the net signed area A between the graph of y=f(x) and the interval [a, b] is

Here is called as the Riemann sum, and the definite integral is Sometimes called the Riemann integral.

Example: Sketch the region whose area is represented by the definite integral, and Evaluate the integral using an appropriate formula from gemoetry. Solution (a): The graph of the integrand is the horizontal line y=3. so the region is a Rectangle of height 3 extending over the interval from 2 to 5. Thus, = the area of the rectangle = 4(3)=12

Example: Sketch the region whose area is represented by the definite integral, and Evaluate the integral using an appropriate formula from gemoetry. Solution (b): The graph of the integrand is the line y=x+2. so the region is a Trapezoid whose base extends from x= -1 to x=1. Thus, Area of trapezoid =1/2(1+3)2=4

Example: Sketch the region whose area is represented by the definite integral, and Evaluate the integral using an appropriate formula from gemoetry. Solution (c): The graph of is the upper semicircle of radius 1, centered At the origin, so the region is the right quarter-circle extending from x=0 to x=1. Thus Area of quarter-circle=

Example: Evaluate Solution: from the figure of y=x-2, we can see that triangular region above and below The x-axis is both 2. Over the interval [0, 4], the net signed area is 4-4=0, and over The interval [0, 2], the net signed area is -2. Thus,

Properties of the Definite Integral Example:

Example: Evaluate Solution: The first integral can be interpreted as the area of a rectangle of height 4 and base 1, So its value is 5, and from previous example, the value of the second integral is Thus,

The Fundamental Theorem of Calculus It is standard to denote the difference F(b) - F(a) as Then (2) can be expressed as We will sometimes write When it is important to emphasize that a and b are value for the variable x.

The Relationship between Definite and Indefinite Integrals For purposes of evaluating a definite integral we can omit the constant of integration in And express as Which relates the definite and indefinite integrals. Example:

Example: (a ) Solution: (b) Solution: (c) Solution:

The FTC can be applied to definite integrals in which the lower limit of integration is Greater than or equal to the upper limit of integration. Example: Solution: Example:

To integrate a continuous function that is defined piecewise on an interval [a, b], split This interval into subintervals at the breakpoints of the function, and integrate Separately over each subinterval in accordance with Theorem Example: Evaluate if Solution:

If f is a continuous function on the interval [a, b], then we define the total area between The curve y=f(x) and the interval [a, b] to be Total area = To compute total area using the above Formula, begin by dividing the interval of Integration into subintervals on which f(x) Does not change sign. On the subintervals for which 0<=f(x), replace |f(x)| by f(x), and on the subintervals for which f(x)<=0 replace |f(x)| by –f(x). Adding the resulting integrals then yields the total area.

Example: Find the total area between the curve and the x-axis over the Interval [0, 2]. Solution: from the graph of, the area is given by