Optimization
Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.
Integration
Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.
Application of Integration
In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.
Volume
In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).
Area
Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.
Find the area bounded by the y2 = 8x and y = 2x.
![Unit 2: AREA BETWEEN TW o CURVES
Consider t he fig ure on
upper curve. over t he interval [a, b], is y, = f (x)
and the
vertical strip. then the sum of the
reclurngles is Ihhere fore Ihe inte gral
the leeft.
Notice that the
l ower curve is y 2 =
g(x). Imagi ne a
are as of t he
y = Rx)
y - Y(x)
("isx) - g(x)]dx
This integral is the area determined by the two
Curves.
We pr esent here t he tw o met hods to find the
area hetween two cuurves
A. Usig the H ORIZONTAL STRIP
From the fig ure on the left, tho area of the strip
is xdy- whor e x - xR - xL- The t otal aroa coan be
found by running this strip starting froom vi goina
dy
to v.
f(y)
Our formula for integration is given bby:
CX. Y
A =
a dy =
IL) dy
Note that xg is the right end of the strip a nd is always n the curve f O). x. is the lleft enc
the stri p and is alwaYs on the curve go). We therefore substitute xR = rO) and x = g0)
prior to integroation.
B. Using the VERTICAL STRIP
we apply the same principle of using
horizontal strip to the vertical strip.
f(x)
From the figure on the left, the area of the
strip is ydx. where y = Yu - YL. The total area
can be found by running this strip starting
from x1 going to x2.
The total area is given by:
| y dz =
/ (yu -– yL) dx
A =
Yu is th e upper end of the strip which is always on the curve (x). yı. is the lower e nd
We th
Here,
of the strip which is always on the curve g(x).
e then Substifute yu - F(x) and y. - g(x).
Puge
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