Determine Area Under Curves

There are various methods for determining the area through curves, in which the most popular method is an anti-derivative method. We can calculate the area by knowing the curve’s equation, the axis enclosing it, and the boundaries. In the general approach, we know the formulas for calculating the area of figures like rectangle, circle, quadrilateral, and square, but there is no such defined formula for determining area through curves. Hereby solving the equation through integration, one can find the area. Let us now see how we can determine the area under different curves.

STEPS TO DETERMINE THE AREA UNDER CURVE

Step I: Find the equation of that curve, axis which encloses the required area and the limits across the area. 

Here equation of the curve is y= f(x)

Step II: Find the integration of the curve

Step III: Apply the limits of the solved integral equation and determine the area under the curve. 

Area = aʃby.dx

= aʃb f(x).dx

= [g(x)]ab

= g(b)- g(a)

METHODS TO DETERMINE THE AREA UNDER CURVE

The method I: In this approach, the required area is broken into many possible small rectangles. Here the summation of all these rectangle areas is the required area under the curve. 

For a curve with equation y= f(x), which is broken into several possible small rectangles with width δx.

Here the limit for the total number of rectangles is up to infinity. 

Area = Lim x→∞ ∑ni=1 f(x). δx

Method II: Here, it is a similar approach as the above method, just the number of rectangles in which the required area is divided is less. Furthermore, the areas of smaller rectangles are added together to get the required area of the curve. We can easily find the area through this method, but its drawback is that it provides an approximate value of the required area. 

Method III: Here, the integration method is used for determining the area through curves. We need the equation of the curve, axis enclosing the curve, and boundaries. 

Area = aʃby.dx

= aʃb f(x).dx

= [g(x)]ab

= g (b) – g (a)

THE FORMULA FOR DETERMINING THE AREA THROUGH CURVES FOR ALL AXES

Area with respect of X-axis: Here the area enclosed by the curve with X-axis and equation y = f(x) is considered. 

Area= a∫b f(X). dX

Area with respect of Y-axis: If the area enclosed is by the curve with equation x= f(y) and with line y = a and y= b, the formula for area is,

Area= a∫b X.dY = a∫b f(Y).dY

Area of the curve below the axis: As a fact, the area below the axis is always negative, so for the area, only the modules are taken.  

Area= |a∫b f(X). dX|

The area below and above the axis: Here, the area above and below the axis is divided into two different areas and is calculated separately. The part which is below the axis and is negative there just the modules are taken. Thus the area is the summation of both areas. 

Area = |A1|+A2

Here formula of the area is,

Area = |a∫bf(X).dX| + b∫c f (X).d(X)

THE FORMULA FOR DETERMINING THE AREA OF SOME SURFACES

• Determining the area of Curve- Circle

It is calculated by calculating the area of the first quadrant of the circle. In that quadrant equation of the curve is y = √ (a2 – x2). Here the limit is from 0 to a, and the area of the circle will be four times the area of the quadrant. So,

Area= 4 0∫a y.dx

=4 0∫a √a2-x2. dx

= 4[x/2√a2–x2 + a2 /2 Sin -1x/a]0a

=4[((a/2) × 0 + (a2/2) Sin-11) – 0]

= 4(a2/2) (π/2) 

= πa2

Therefore, the Area of Circle= πa2 square units.

• Determining the area under the curve- Parabola

In the case of the parabola, it has two symmetric parts. If we take a parabola that is symmetric to the X-axis, then the equation of the curve is y2 = 4ax. Here we will find the area of the first quadrant with limits 0 to a. And then, to find the area of the parabola, we will double it. So, the formula for the area of the parabola is,

Area = 20∫a √4ax.dx

=4√a∫0 √x.dx

=4√a [(2/3. a3/2)] 0a

=4√a [(2/3. a3/20-0]

= 8a2/3

Hence, the area enclosed by the parabola= 8a2/3 square units.

• Determining the area under curve- Ellipse

In the case of Ellipse, we calculate the area of one quadrant and then multiply it by four to get the area of Ellipse. Here the equation of the curve with major and minor axis 2a and 2b respectively is y= b/a.√ (a2 – x2). The boundary limit for integration of the equation is from 0 to a. So, the formula of area of Ellipse is, 

Area = 40∫a y.dx

=40∫a b/a.√ (a2 – x2). dx

=4b/a [x/2. √a2−x2 +a2/2Sin−1x/a] a0

=4b/a. a2/2. π/2

=πab

Hence, the Area of Ellipse= πab square units

CONCLUSION

Determining the area of the irregular plane surface is difficult, and in finding antiderivatives, the method is very useful. Through the above methods, we understand how to find the area under a curve with boundary limits and enclosed axis.