Chapter:8

What are Applications of the Integrals ?

  • Applications of the integrals means ‘Use of integrals’.
  • For finding the area of any curve we must have knowledge of integrals and conic section.

Area Under Simple Curve

We consider the easy and intuitive way to finding the area bounded by the curve. From the figure ,we think of area under the curve as composed of large number of very thin vertical strip (black color strip). And we consider this black vertical strip as a rectangle .

Case 1:

The area bounded by the curve y = f(x), the x-axis and the ordinates
x = a and x = b is given by (above figure) As we know area of rectangle ‘A’= length x breath

dA = y dx
⇒ dA = f(x)dx

AREA UNDER SIMPLE CURVE
AREA UNDER SIMPLE CURVE

Case 2:

The area bounded by the curve x = f(y), the y-axis and the abscissa
y = c and y = d is given by Similarly, in case 1

area bounded
abscissa

Case 3:

If the curve y = f(x) lies below the x−axis, then the area bounded by the curve y = f(x), the x−axis and the lines x = a and x = b come out to be negative. Thus , if the area is negative (Area never negative), we take its absolute value, i.e.;

the curve y
the curve

Area Enclosed By The Parabola (X2 = 4ay) And Its Latus Rectum

In Case Of Horizontal Strip:

In Case Of Horizontal Strip
AREA ENCLOSED BY THE PARABOLA

Area Enclosed By The Ellipse (x2/a2+y2/b2=1)

In Case Of Horizontal Strip:

In Case Of Horizontal Strip:
AREA ENCLOSED BY THE Ellipse

Area Enclosed By The Circle (X2 + Y2 = a2)

In Case Of Horizontal Strip:

AREA ENCLOSED BY THE CIRCLE
AREA ENCLOSED BY THE CIRCLE