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MAT 212 Brief Calculus Section 5.4 The Definite Integral

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The Definite Integral Given a function f that is continuous on the interval [a,b], if we divide the interval into n subintervals and choose a point, in each interval, the definite integral from a to b is said to be a and b are our limits of integration

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The Definite Integral Given a function f that is continuous on the interval [a,b], if we divide the interval into n subintervals and choose a point, in each interval, the definite integral from a to b is said to be f(x) is called our integrand

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The Definite Integral Given a function f that is continuous on the interval [a,b], if we divide the interval into n subintervals and choose a point, in each interval, the definite integral from a to b is said to be dx represents our rectangle widths and tells us what our variable of integration is

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The Definite Integral Given a function f that is continuous on the interval [a,b], if we divide the interval into n subintervals and choose a point, in each interval, the definite integral from a to b is said to be The definite integral calculates the net area underneath f(x) and above the x-axis between a and b

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The Definite Integral Given a function f that is continuous on the interval [a,b], if we divide the interval into n subintervals and choose a point, in each interval, the definite integral from a to b is said to be This can be thought of as the calculation of the accumulation of f

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The Definite Integral Given a function f that is continuous on the interval [a,b], if we divide the interval into n subintervals and choose a point, in each interval, the definite integral from a to b is said to be We saw that we can calculate a definite integral using the fundamental theorem of calculus, let’s take a quick look at the fundamental theorem

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If f is continuous on the interval [a,b], and f(x) = F ‘(x), then To find, first find F, then calculate F(b) - F(a) This method of computing definite integrals gives an exact answer.

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Examples

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Notation for the general antiderivative of a function f(x) looks like the definite integral without the limits of integration. Definite: General:

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Definite: General: This gives you a number This represents a family of functions

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The Definite Integral If a, b, and c are any numbers and f is a continuous function, then

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The Definite Integral Let f and g be continuous functions and let c be a constant, then

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The Definite Integral Let f and g be continuous functions and let c be a constant, then

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Find the exact value of the area between by the graphs of y = e x + 3 and y = x + 2 for 0 ≤ x ≤ 2

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Find the exact value of the area between by the graphs of y = x + 1 and y = 7 - x for 0 ≤ x ≤ 4

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Find the exact value of the area enclosed by the graphs of y = x 2 and y = 2 - x 2 Let’s look at #13 from the book

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