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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

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

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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

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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

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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=

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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,

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Properties of the Definite Integral Example:

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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,

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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.

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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:

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

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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:

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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 6.5.5. Example: Evaluate if Solution:

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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.

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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

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