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| Welcome to our last and final topic of this semester of Calculus. It is called the antiderivative or integral. Its function is to find the area under the curve, just as the derivative finds the slope of the curve. |
| The Indefinite Integral: Notice the notation of integrals. it is similar to that of derivatives. The way to view this is that the f(x) is the height and the dx is a piece of the x. So when you integrate it you get a piece of the area. And what that curvey line is is an big "S" which means sum. So what we are doing is summing up the pieces of the areas to get the whole area. That is it. |
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| The indefinite integral means that you are finding the generic formula for the area so that you can find the area inbetween anytwo points. This will help you to visualize the summing up this that I was talking about before. Something to notice is that as you increase the number of boxes, the closer it gets to the real sum of the area. with(student): middlebox(x^4*ln(x), x=2..4, color=YELLOW,title="4 Boxes"); middlebox(x^4*ln(x), x=2..4, 12, color=BLUE,title="12 Boxes"); middlebox(x^4*ln(x), x=2..4, 100, color=RED,title="100 Boxes"); |
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| Now what you do with an integral is the same as what you do for a derivative except for backwords. So the derivative of sin(x)=cos(x), so the antiderivative of cos(x)=sin(x)+C. Now where the heck did that C come from? Well, it doesn't matter where it came from, put it regardless. This is the same for polynomials also! DON'T FORGET THE C CONSTANT, UNLIKE MAPLE! |
| Int(cos(x),x):=int(cos(x),x); |
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| The Definite Integral |
| The Definite Integral is the same as the Indefinite Integral with the exception of the fact that you plug in the bounds or limits. It always goes upper minus lower. So if we have the function f(x) and the antiderivative of that is F(x)+C then the area between the points a and b on the x-axis is equal to F(b)-F(a) |
| withstudent: Int(f(x),x=a..b):=F(b)-F(a); |
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| Here is a simple sample of that. Int(cos(x),x=0..Pi/2):=int(cos(x),x=0..Pi/2); |
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| That is it for integrals. Have fun plugging away at the stupid trapezoid rule and its companions. Besides from this its just memorizing tedious little rules. Good Luck. May you have good luck in your math career ahead of you! |
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