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6.3 Definite Integrals and the Fundamental Theorem

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We have seen that the area under the graph of a continuous nonnegative function f(x) from a to b is the limiting value of Riemann sums of the form f(x1)Δx + f(x2) Δx +…+f(xn) Δx as the number of subintervals increases without bound, or as Δx approaches zero.

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It can be shown that even if f(x) has negative values, the Riemann sums still approach a limiting value as Δx approaches zero. This number is called the definite integral of f(x) from a to b and is denoted by

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That is We know that the Riemann sum on the right side approaches the area under the graph of f(x) from a to b. So, the definite integral of a nonnegative function f(x) equals the area under the graph of f(x).

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By geometry So

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If f(x) is negative at some points in a given interval, we can still give a geometric interpretation of the definite integral. Suppose we have the following graph of a function f(x) with the interval from a to b

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The Riemann sum is equal to the area of the rectangles above the x-axis minus the area of the rectangles below the x-axis. Note: As Δx approaches zero, the Riemann sum approaches the definite integral.

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The rectangular approximations approach the areas bounded by the graph that is above the x-axis minus the area bounded by the graph that is below the x-axis. This gives us the following geometric interpretation of the definite integral…

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Suppose that f(x) is continuous on the interval Then is equal to the area above the x-axis bounded by the graph of y = f(x) from x = a to x = b minus the area below the x-axis bounded by y = f(x).

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We have Considering this figure

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From the geometry, we see

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We are now ready for the theorem that indicates how to use antiderivatives to compute the definite integral…

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Fundamental Theorem of Calculus Suppose that f(x) is continuous on the interval and let F(x) be an antiderivative of f(x). Then This theorem connects the two key concepts of calculus – the integral and the derivative.

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F(b) – F(a) is called the net change of F(x) from a to b. It is represented symbolically by

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Use the fundamental theorem of calculus to evaluate the following definite integrals

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