Class 12 Mathematics — Chapter 2: APPLICATION OF INTEGRALS

What you'll learn

• ▸Area as a Definite Integral (Vertical Strips)
• ▸Area with Respect to the y-axis (Horizontal Strips)
• ▸Area of Regions Below the x-axis
• ▸Area of Composite Regions
• ▸Area of a Circle using Integration
• ▸Area of an Ellipse using Integration
• ▸Area under Trigonometric Curves
• ▸Area involving Absolute Value Functions

Chapter Summary

--- PAGE 1 --- 12080CH08 Chapter 8 APPLICATION OF INTEGRALS ❖ One should study Mathematics because it is only through Mathematics that nature can be conceived in harmonious form. – BIRKHOFF ❖ 8.1 Introduction In geometry, we have learnt formulae to calculate areas of various geometrical figures including triangles, rectangles, trapezias and circles. Such formulae are fundamental in the applications of mathematics to many real life problems. The formulae of elementary geometry allow us to calculate areas of many simple figures. However, they are inadequate for calculating the areas enclosed by curves. For that we shall need some concepts of Integral Calculus. In the previous chapter, we have studied to find the area bounded by the curve y = f(x), the ordinates x = a, x = b and x-axis, while calculating definite integral as the limit of a sum. Here, in this chapter, we shall study a specific application of integrals to find the area under simple curves, area between lines and arcs of…

Practice Questions from this Chapter

Tap "Get Solution" on any question to ask our AI tutor.

1. Show real-world integral examples. Get Solution →
2. Visualize integral area calculations. Get Solution →
3. Explain calculus in simple terms. Get Solution →
4. What is the formula for the area of an elementary vertical strip under the curve y = f(x)? Get Solution →
5. How is the total area A of a region under y=f(x) from x=a to x=b expressed symbolically? Get Solution →
6. What is the formula for the area bounded by the curve x=g(y), the y-axis, and lines y=c and y=d? Get Solution →
7. If the calculated area for a region that lies entirely below the x-axis is negative, what value should be taken? Get Solution →
8. What is the final formula for the area of a circle with radius 'a', as derived in Example 1? Get Solution →

Did you know?

• 💡 Ancient Egyptians used geometry to survey land after Nile floods.
• 💡 Circles enclose the maximum area for any given perimeter length.
• 💡 The Babylonians calculated circle areas using an early approximation of pi.
• 💡 Calculus can find the exact area of incredibly complex and curved shapes.
• 💡 Engineers use integrals to calculate how much material is needed for curved structures.

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