Transcript Riemann Integral and it’s everyday use.

Riemann Integral and
it’s everyday use.
Prepared by: Cherry Sudartono
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
Who Formulated Integration?
Sir Isaac Newton
Gottfried Wilhelm Leibinz
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.
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.
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.
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
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 ∑
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 →∞
=│