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Integration





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Integration is the measure of the area under a curve. What it does is to take the sum of rectangles under the curve. As more rectangles are inscribed, but with smaller area, we get a better approximation. I don't have a picture of this particular item to show you, but I have something that will illustrate my point.




You can see that for more triangles that are in that circle, the closer to the area of the circle the inscribed polygon is. This is also true for inscribed rectangles under any curve drawn in the cartesian coordinates. So, the integration is the sum of rectangles inscribed under any graph, where each rectangle is f(xn)(xn-xn-1) for any point along the curve. As I said earlier, integration is an inverse function of derivatives, so they're somewhat related. The symbol for integration is a flattened capital S, which stands for summa in Greek. The integration of f(x) is written as f(x) dx, where dx is the difference between xn and xn-1. See, the examples below.

If f(x) = xn, then f(x) dx = xn+1/(n+1).
if f(x) = cxn, then f(x) dx = cxn+1/(n+1)
.

Some other integration formulas:

If f(x) = ex, then f(x) dx = ex
If f(x) = sin x, then f(x) dx = -cos
.If f(x) = sin cx, then f(x) dx = (-cos cx)/c
If f(x) = cos x, then f(x) dx = sin x.
If f(x) = cos cx, then f(x) dx= (sin cx)/c..



Integral Quiz
1. If f(x) = x4, what is f(x) dx?
2. If f(x) = cos 4x, what is f(x) dx?
3. If f(x) = ex, what is f(x) dx?
4. What does the integral measure?




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