Definite Integral Across Addition, SubtractionDate: 8/10/96 at 15:8:37 From: Anonymous Subject: Definite Integral Across Addition, Subtraction What is the integral of 3(x^2)-5x+9 from 0 to 7? Date: 8/11/96 at 18:32:14 From: Doctor Paul Subject: Re: Definite Integral Across Addition, Subtraction We first need to know a property of integrals: / / / | [ a(x) + b(x) ] dx = | a(x) dx + | b(x) dx / / / So we can rewrite your integral from: /7 | 3x^2 - 5x + 9 dx /0 to: / / / | 3x^2 dx + | 5x dx + | 9 dx / / / / / now remember that | a * b(x) dx = a | b(x) dx / / so let's pull the constants out in front: / / / 3 | x^2 dx + 5 | x dx + 9 | dx / / / ^^^^^ note that we really have x^0 here.. Now you need to know how to integrate: If you integrate x^n dx then you get [ (1/(n+1)) * x^(n+1) ] Let's perform the integration: 3*[ (1/3) * x^3 ] + 5*[ (1/2) * x^2) ] + 9*[ (1/1) * x^1) ] Simplify it out.. x^3 + (5/2)x^2 + 9x since we're looking for a definite integral we don't need to add a constant of integration (the '+ C' at the end) let's apply the limits: |7 x^3 + (5/2)x^2 + 9x| |0 Now we use the Fundamental Theorem of Calculus to solve this.. /b | f(x) dx = F(b) - F(a) where F(x) is *any* antiderivitive of f(x) /a The word *any* in the definition above allows us to choose our constant of integration to be zero in these definite integral problems. Let's solve it. We already know F(x). F(x) = x^3 + (5/2)x^2 + 9x F(7) - F(0) = 283.5 - 0 = 283.5 I hope this helps clear things up.. Regards, -Doctor Paul, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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