Class 12 Mathematics Chapter 8: Application of Integrals

CBSE 2026–27 | Detailed NCERT-aligned study resource

This chapter turns definite integration into a geometry tool. The key skill is not merely evaluating an integral; it is correctly identifying the region, finding intersections, choosing limits, and deciding whether to integrate with respect to x or y.

1. Area Under a Curve

If a continuous curve y=f(x) lies above the x-axis from x=a to x=b, its area with the x-axis is ∫ₐᵇ f(x)dx.

Worked Example 1

Find the area under y=x² from x=0 to x=2.

A=∫₀²x²dx=[x³/3]₀²=8/3. Therefore, the area is 8/3 square units.

2. Area When the Curve Is Below the x-axis

A definite integral may be negative because it represents signed area. Geometric area is always non-negative. If f(x)<0 throughout the interval, area is −∫ₐᵇf(x)dx.

Example

For y=−x² from 0 to 1, the integral is −1/3, but the geometric area is 1/3 square unit.

3. Area Between Two Curves

When two curves y=f(x) and y=g(x) enclose a region, first solve f(x)=g(x) to obtain intersection points. Then identify which curve is above the other.

Worked Example 2: y=x and y=x²

Intersections satisfy x=x², so x=0 and x=1. Between 0 and 1, x is above x².

A=∫₀¹(x−x²)dx=[x²/2−x³/3]₀¹=1/2−1/3=1/6 square unit.

4. Area With Respect to y

Sometimes horizontal strips are simpler. If x=right boundary and x=left boundary, then A=∫(right−left)dy. Always sketch first.

5. Standard Curves in the CBSE Scope

Circle

For x²+y²=a², the upper semicircle is y=√(a²−x²). By symmetry, the area of the complete circle can be obtained by integrating the upper half and doubling it.

Parabola

For y²=4ax, write y=2√(ax) for the upper branch when appropriate. The limits must come from the geometry of the bounded region.

Ellipse

For x²/a²+y²/b²=1, the positive upper branch is y=b√(1−x²/a²). Symmetry can reduce the required integral.

6. How to Solve an Area Question

1. Draw a rough labelled graph.
2. Find intersection points.
3. Decide whether vertical or horizontal strips are easier.
4. Identify upper/lower or right/left boundaries.
5. Write the limits.
6. Set up the definite integral.
7. Evaluate and state square units.

7. Exam-Level Example

Find the area enclosed between y=x and y=4−x². First solve x=4−x², giving x²+x−4=0. The roots provide the limits. On the interval between the intersections, determine the upper curve before writing ∫(4−x²−x)dx. The essential scoring steps are the intersection calculation, correct ordering of curves, limits, integration and final area.

8. Common Mistakes

• Writing limits without finding intersections.
• Subtracting lower curve from upper curve in the wrong order.
• Forgetting that geometric area cannot be negative.
• Using dx automatically when dy gives a simpler region.
• Failing to split the integral when the upper/lower curve changes.

9. Practice Set

1. Find the area between y=x and y=x².
2. Find the area between y=4−x² and the x-axis.
3. Find the area enclosed by a circle and a coordinate axis using symmetry.
4. Find the area enclosed by a suitable parabola and a line after determining their intersections.
5. For an ellipse in standard form, set up an integral for one quadrant and use symmetry.
6. Draw the region bounded by two given curves and decide whether dx or dy is more efficient.

10. Quick Revision