Application of Integrals

Applications of Integrals – Description (Class 12)

Applications of Integrals deal with the use of definite integrals to find areas of plane regions bounded by curves, straight lines, and the coordinate axes. This chapter extends the concept of integration beyond algebraic evaluation to geometrical interpretation.


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1. Area Under a Curve

If a function is continuous and non-negative on the interval , then the area bounded by the curve, the x-axis, and the ordinates and is given by:

\text{Area} = \int_a^b f(x)\,dx


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2. Area Above and Below the x-Axis

Area above the x-axis is taken as positive.

Area below the x-axis is taken as negative.

To find the actual area, the modulus of the integral is taken.



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3. Area Between Two Curves

If two curves and () intersect at and , then the area enclosed between them is:

\text{Area} = \int_a^b [f(x) - g(x)]\,dx


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4. Area Bounded by Curves and Axes

Integrals can also be used to find areas enclosed by:

A curve and the x-axis

A curve and the y-axis

Two curves and a line


In such cases, limits of integration are chosen according to the points of intersection.


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5. Area with Respect to y-Axis

If the equation is in the form , then the area is calculated as:

\text{Area} = \int_c^d f(y)\,dy


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6. Practical Importance

Applications of integrals are used in:

Finding areas of irregular shapes

Solving problems in physics, engineering, economics, and statistics

Calculating distance, work done, and average values



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Conclusion

The chapter Applications of Integrals helps in understanding how integration is applied to real-life and geometrical problems, especially in finding areas bounded by curves. It forms a strong foundation for advanced applications of calculus.


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