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Riemann Integral and it’s everyday use. Prepared by: Cherry Sudartono.

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1 Riemann Integral and it’s everyday use. Prepared by: Cherry Sudartono

2 I Integration is a core concept of advanced mathematics, specifically in the fields of calculus and mathematical analysis. What is Integral? Given a function f(x)of a real variable x and an interval (a,b), the integral is related to the area of a region bounded by the graph of f, the x-axis, & the vertical lines x = a & x = b

3 Who Formulated Integration? Sir Isaac NewtonGottfried Wilhelm Leibinz

4 Riemann Integral Bernhard Riemann cultivated a new method called the Riemann Integral which approximates the area of a curvilinear area by breaking the area into thin vertical blocks, like shown on the image.

5 Using Riemann Integral In Physics We can use Riemann Integrall to find the distance traveled by an object if we know the velocity of the whole journey and the amount of time. We can easily retrieved this information from the Velocity versus Time graph. The “area under the curve” is actually the distance traveled.

6 Distance Traveled by a Boeing I travel back to my hometown with a Boeing 747. It usually is a 20 hours flight with the top velocity of the aircraft being approximately 560 miles/hours.

7 1 st Step of Riemann Integral – Finding the area of slice To find the Area of the slice, we assume that the width of the slice is  x. As for the height, since the height still resides in the graph y = f(x), we consider the height to be f(x). Therefore, the formula for the area of the slice, A = f(x)*  x. f(x) = (5/3)x 2 + (9/7)x + 23

8 2 nd Step – Configuring Riemann Sum The next step of the Riemann Integral is to configure the formula of the Area of the slice into “infinite slices”. Basically we’re looking for the sum of all of the Area in infinite form. The basic formula will look something like this. In our case however, it’ll look like this ∑

9 3rd Step – Step to the Limit In this step, we have to make sure that we configure the Sum of the infinite slice and limit it to infinity. At the same time we are integrating the formula of the infinite slice. Note also that we let  x to be 0 therefore  x becomes dx. Area = ∫ f(x) dx lim →∞ = │

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