Application of Integrals: Overview - JEE PDF Download

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Integrals have their application in both science and maths.
In maths, the application of integral is made to determine the area under a curve, the area between two curves, the center of mass of a body, and so on.
Whereas in science (Physics in particular), the application of integrals is made to calculate the Centre of Gravity, Mass, Momentum, Work done, Kinetic Energy, Velocity, Trajectory, and Thrust.

Area Under Curve

• Area bounded by curve y = f(x), the ordinates x = a, x = b and x–axis is given by
  
• Area bounded by curve x = f(y), the abscissae y = c, y = d and y-axis is given by  
• Area bounded by the curves  y = f(x) and y = g(x) is given by
  where a and b are the ordinates of points of intersection

Illustration: Determine the area enclosed by the circle x² + y² = a².
Solution: Given, circle equation is x² + y² = a².

From the given figure, the whole area enclosed by the given circle = 4(Area of the region bounded by the curve AOBA, coordinates x=0 and x=a, and the x-axis)
As the circle is symmetric about both the x-axis and y-axis, the equation can be written as
...(1)

From the given circle equation, y = ±√(a²-x²)

As the region, AOBA lies in the first quadrant of the circle, we can take y as positive.
∴ y =√(a2-x2)

On Substituting y = √(a²-x²) in equation (1),

On Integrating,

Hence, the area enclosed by the circle x² +y² =a²  is πa².