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

2. What is Integral?
Integration is a core concept of advanced mathematics, specifically in the fields of calculus and mathematical analysis.
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? Gottfried Wilhelm Leibinz
The principles of integration were formulated by Sir Isaac Newton and Gottfried Wlhelm Leibinz in the late seventeenth century. Though the fundamental theorem of calculus which they developed independently, integration is connected with differentiation, and the definite integral of a function can be easily computed once an antiderivative is known.
Sir Isaac Newton
Gottfried Wilhelm Leibinz

4. Riemann Integral
The basic idea of the Riemann Integral is to use very simple approximations for the area. By taking better and better approximations, we can say that “in the limit” we get exactly the area under the curve.
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 747-400
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. 1st 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)x2 + (9/7)x + 23

8. 2nd 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 →∞
= │