Calculate the area of ​​an object using the integral method

A problem that we often face is if we are faced with the calculation of the building area that is limited not by a straight line. An example is how we calculate the area of ​​a wake in Figure 1 below,

To find the building area approach is to make plots in the form of a unit square as shown in Figure 1. If we consider the area of ​​the inner and outer areas of the building, the building area is between 12 and 16 area units.

Furthermore, the area of ​​a closed curve in the Cartesian plane will be studied. Figure 2 shows an area limited by the curve y = f(x), X axis, line x = a, and x = b.


To obtain a broad approach to the area, the method is to make several vertical lines (strips) such that they form a rectangle. The total area of ​​the rectangle is the area wide approach.
Suppose that one of the rectangular pieces is named PP‘Q’Q with the coordinates of point P (x, y). The rectangle width P'Q' we call akanx, and the area of ​​PP‘Q’Q we call δL. The area of ​​a rectangle is the length of the width, then δL = y δx

The number of rectangular areas from x = a to x = b can be expressed with
With n is a lot of rectangles. The total area calculation will be accurate if the selected δx is very small to near zero, that is, in a limit. So,

We can write the form above as an integral form below,

Our next question is whether the area in question is an anti-derivative of a function? To answer these questions, will be explained in the next chapter.

Example 1.
Show with an area shading that is formulated with an integral form, below

Answer:
First draw each curve whose equation is y = 2x + 3 and y = x2, then make the shading as mentioned above.

Fundamental Theorem

Theorem: If f is a continuous function in the intervals [a, b] and F is anti derivative f on [a, b], then

Proof of the function f (x) in Intergral Fundamental is as follows,

The shaded area in Figure 5 is the area bounded by the curve y = f (x), X axis, line x = a, and x = b. Suppose the rectangle width in the image is h, then the area of ​​PP1Q1Q is

L(x + h) = L(x)
And this area of ​​magnitude lies between the area of ​​a small rectangle and a large rectangle, then
For h à 0 then,


The area of ​​x = a to x = b is


Certain Intergral

The form of writing F(b) - F(a) on the Fundamental Theorem is usually expressed by. Writing in full is as follows:
Note: The integral written in the notation is called the Specific Integral because the result is a certain value, a is called the lower limit and b is called the integral upper limit.

Example 2.

Calculate Integral below,


Example 3.
Calculate Integral below,


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