Class 12 Notes - Application of Integrals

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Application of Integrals

The application of integrals is a crucial topic in Class 12 mathematics, focusing on how definite integrals can be used to solve real-world problems. The most common applications include finding the area under curves, the area between two curves, and solving problems related to physics and engineering.

1. Area Under a Curve

The area bounded by a curve y = f(x), the x-axis, and the lines x = a and x = b is given by:

Area = ∫ab f(x) dx

This integral gives the total area between the curve and the x-axis from x = a to x = b.

2. Area Between Two Curves

If you have two curves y = f(x) and y = g(x) (where f(x) ≥ g(x) in [a, b]), the area between them is:

Area = ∫ab [f(x) - g(x)] dx

3. Steps to Find Area Between Curves

  1. Sketch the curves to identify the region whose area is to be found.
  2. Find the points of intersection (limits of integration).
  3. Set up the definite integral using the correct upper and lower functions.
  4. Evaluate the integral to get the area.

4. Example Problems

Example 1: Find the area bounded by the curve y = x2, the x-axis, and the lines x = 1 and x = 3.

Solution:

  • Area = ∫13 x2 dx = [x3/3]13 = (27/3) - (1/3) = 9 - 1/3 = 8⅔ units²

Example 2: Find the area enclosed between y = x and y = x2 from x = 0 to x = 1.

Solution:

  • Area = ∫01 (x - x2) dx = [x2/2 - x3/3]01 = (1/2 - 1/3) = 1/6 units²

5. Applications in Physics and Engineering

  • Finding distance, area, and displacement in physics problems.
  • Calculating work done by a variable force.
  • Determining the center of mass, volume, and surface area in engineering.

6. Practice Questions

  1. Find the area bounded by y = 2x + 1, the x-axis, x = 0, and x = 2.
  2. Calculate the area between y = sin x and y = cos x from x = 0 to x = π/2.
  3. Find the area enclosed between y = ex and y = e-x from x = 0 to x = 1.

7. Summary

  • Definite integrals are used to find areas under curves and between curves.
  • Always sketch the region and identify limits before integrating.
  • Applications extend to physics, engineering, and real-life problems.