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Calculus II Notes 5.2

# Calculus II Notes 5.2 - Calculus II-Stewart Dr Berg Spring...

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Calculus II- Stewart Dr. Berg Spring 2010 Page 1 5.2 5.2 The Definite Integral We saw in the previous section that taking limits of estimating sums can be used to find area and distance. Now we generalize the ideas involved. Definition If f is a function defined on the interval [ a , b ] , we divide the interval into n subintervals of equal width Δ x = ( b a )/ n . Now let x 0 = a , x 1 , x 2 , , x n = b be the endpoints of these subintervals, and let x 1 *, x 2 *, , x n * be any sample points in these subintervals (so x i * lies somewhere in the i th subinterval). Then the definite integral of f from a to b is f ( x ) dx a b = lim n →∞ f ( x i *) Δ x i = 1 n if this limit exists. If it does exist, we say that f is integrable on [ a , b ] . Note: I n = f ( x i *) Δ x i = 1 n is called a Riemann sum and the convergence of the integral is the convergence of the sequence of Riemann sums, and since each Riemann sum is a number, the integral is a number. In the notation f ( x ) dx a b the integrand is f ( x ), a is the lower limit , and b is the upper limit . Notice also that, since we do not demand that the function be positive, the integral could be a negative number. This makes sense when calculating a net electric charge, which could be positive or negative, or when calculating distance, etc. Theorem If f is continuous on [ a , b ] or has at most a finite number of jump discontinuities, then f is integrable on [ a , b ] . As we have seen, for the sake of simplicity in evaluating integrals without using the fundamental theorem, we often use a regular partition and the right endpoint of each subinterval.

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