Mathematics Class 12 - Application Of-Integrals Notes

Application of Integrals

Welcome to the chapter on Application of Integrals for Class 12. In this chapter, you will learn how to use definite integrals to find areas under curves, between curves, and solve real-life problems involving integration. By the end of this chapter, you will be able to apply integrals to geometry and practical situations.

Key Concepts

• Definite Integral: Represents the area under a curve between two points.
• Area under a Curve: The region between the curve and the x-axis.
• Area between Curves: The region between two curves over a given interval.

Area under a Curve

The area under the curve y = f(x) from x = a to x = b is given by:

Area = ∫_a^b f(x) dx

Example: Find the area under y = x² from x = 0 to x = 2.
Area = ∫₀² x² dx = [x³/3]₀² = (8/3) - 0 = 8/3 units²

Area between Two Curves

The area between two curves y = f(x) and y = g(x) from x = a to x = b is:

Area = ∫_a^b [f(x) - g(x)] dx, where f(x) ≥ g(x) in [a, b].

Example: Find the area between y = x² and y = x from x = 0 to x = 1.
Area = ∫₀¹ (x - x²) dx = [x²/2 - x³/3]₀¹ = (1/2 - 1/3) = 1/6 units²

Properties and Applications

• Areas can be found for regions bounded by curves, lines, and axes.
• Integrals are used in physics, engineering, economics, and biology to find quantities like distance, area, and total value.
• If the curves intersect, find the points of intersection to set the limits of integration.

Practice Questions

1. Find the area under y = 3x from x = 1 to x = 4.
2. Calculate the area between y = x and y = 2x - x² from x = 0 to x = 1.
3. If y = f(x) is below the x-axis, what does the definite integral represent?
4. Find the area bounded by y = x², y = 4, x = 0, and x = 2.
5. Explain how to find the area between two intersecting curves.

Challenge Yourself

• Find the area enclosed between y = sin x and the x-axis from x = 0 to x = π.
• Calculate the area between y = e^x and y = e^-x from x = 0 to x = 1.

Did You Know?

• The concept of integration was developed by Isaac Newton and Gottfried Wilhelm Leibniz.
• Integrals are used to find volumes, surface areas, and many other quantities in science and engineering.

Glossary

• Definite Integral: The value of the integral with upper and lower limits, representing area.
• Curve: A line that is not straight.
• Intersection: The point where two curves meet.

Answers to Practice Questions

1. Area = ∫₁⁴ 3x dx = [3x²/2]₁⁴ = (3×16/2) - (3×1/2) = 24 - 1.5 = 22.5 units²
2. Area = ∫₀¹ [(2x - x²) - x] dx = ∫₀¹ (x - x²) dx = [x²/2 - x³/3]₀¹ = (1/2 - 1/3) = 1/6 units²
3. It represents the area below the x-axis, which is taken as negative. For total area, take the absolute value.
4. Area = ∫₀² (4 - x²) dx = [4x - x³/3]₀² = (8 - 8/3) - 0 = 16/3 units²
5. Find the points of intersection, set them as limits, and integrate the difference of the functions over that interval.

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Practice applying integrals to find areas and solve real-world problems!