Lesson 18a - Integral Test and P-test

Lesson 18a - Integral Test and P-test - IntegralsandAreas b...

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Integrals and Areas Recall that the major application of an integral is finding the area under the curve. is could be expressed as: b x This could be expressed as: a dx x f ) (
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Integrals and Areas We know that if we take a summation rather than an integral, we are approximating the area rather than finding the full area. Caution: The integral will not be the value of the sum as we are taking boxes of width 1 (not a finely efined area as done by an integral) defined area as done by an integral). Consider a graph that is strictly decreasing , positive , and continuous . We can take ti l b hi th idth f b t b 1 d hih t t b th a partial area by choosing the width of our box to be 1, and our height to be the value of the function: This means that a 2 = f(2), a 3 = f(3) …
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Integrals and Areas We know that the area of the box a n (which will be f(n)) will be smaller than the value of the integral: is means that adding the areas: This means that adding the areas: n n dn n f a a a a 1 4 3 2 ) ( ... This is almost the series, but we are missing the first term: n n dn n f a a a a a a 1 1 4 3 2 1 ) ( ...
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Integrals and Areas n This means that our partial sum from 1 to n is: n dn n f a a a a a a 1 1 4 3 2 1 ) ( ... n n dn n f a S 1 1 ) ( If the integral is continuous and converges to a number, this means we have (going to infinity) n m Since the numbers are strictly decreasing, and are positive, we have that the Sums i t t l d l bt bd d b b L dn n f a S n n 1 1 ) ( lim S n
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Lesson 18a - Integral Test and P-test - IntegralsandAreas b...

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