Integral Integrals for the CSET

Feb
18

Filed Under CSET Math |

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Derivatives take us from a function to an expression, integrals take us the opposite direction– from an expression to a function.

Riemann Sums

A Riemann sum is an approximation for the area beneath a curve that is achieved using rectangles.

In the graph below of y = x2 + 1, the interval [0, 3] is shown. We want to figure out the area of the yellow area. We will use rectangles to approximate the shaded area.

Right SumTo approximate the area on the x-interval [0, 3], we will use three rectangles each with a width of 1.The right sum method means that the rectangles will be the height of the function at the right side of each interval as in the
figure below:

We can approximate the area beneath the curve by adding the areas of the three rectangles. The area of a rectangle is equal to length times width, therefore 1 · 2 + 1 · 5 + 1 · 10 = 17. So the right Riemann
approximation with n = 3 rectangles is 17.Left SumThe left sum method means that the rectangles will be the height of the function at the left side of each interval as in the figure below:

Note that as n (number of intervals or rectangles) increases, the approximate area gets more accurate.What is the integral of “one over cabin” with respect to “cabin”?
Answer: Natural log cabin + c = houseboat.

By Joe Kodiar
http://www.ACEtheCSET.com
Joe Kodiar Integral Integrals for the CSET

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