medium difficulty10 min

Area Between Two Curves

The area between two curves y = f(x) and y = g(x) from x = a to x = b is ∫_a^b [upper curve - lower curve] dx, provided the same curve stays above on the whole interval.

In one line: Between two curves means big boundary minus small boundary in the strip direction.

Practice This Concept

Understand it

This concept applies when a closed region is formed by two curves or by curves along with given boundaries.
The first task is to find intersection points, because they often become limits.
After that, decide which curve is above the other on the interval.
If the curves cross inside the interval, split the integral at the crossing point.

Example

The area between y = x and y = x² from x = 0 to x = 1 is ∫₀¹ (x - x²) dx = [x²/2 - x³/3]₀¹ = 1/2 - 1/3 = 1/6 square unit.

Formula and rule

Key rule: Area = ∫_a^b [f(x) - g(x)] dx when f(x) ≥ g(x) on [a, b].

If using horizontal strips, Area = ∫_c^d [right curve - left curve] dy.

Conditions: limits must match the bounded region, intersections must be found when not given, and the order must be changed or split if the curves cross.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Find the area enclosed between y = x and y = x².

1Intersections: x = x² gives x(x - 1) = 0, so x = 0 and x = 1.
2On (0, 1), x > x², so upper curve is y = x and lower curve is y = x².
3Area = ∫₀¹ (x - x²) dx = [x²/2 - x³/3]₀¹ = 1/2 - 1/3 = 1/6 square unit.

Method you can reuse

1Write equations of both curves.
2Solve them simultaneously to find intersection points.
3Sketch both curves and shade the enclosed region.
4Decide vertical or horizontal strips.
5For dx, identify top minus bottom; for dy, identify right minus left.
6Check whether the order stays the same over the interval.
7Integrate, substitute limits, simplify and write square units.

Sketch it in your answer

Draw both curves on the same axes, mark their intersection points, shade only the enclosed region, and label the upper and lower curve on each interval. If using dy, label right and left boundaries instead.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

For y = x and y = x² on [0, 1], which curve is above?

Think first, then show answer

y = x is above y = x² on [0, 1], because for 0 < x < 1, x > x².

Common wrong answer

For y = x and y = x², writing ∫₀¹ (x² - x) dx gives -1/6, which is signed area and cannot be the final area of a region.

Don't mix this up

Many students subtract in the order written in the question instead of subtracting lower curve from upper curve.

Exam tip

After finding intersection points, test one x-value between the limits to decide the top curve. This small check prevents the most common sign error.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Often asked as a long answer problem where students must solve intersections, sketch the region, choose the correct subtraction order, integrate a polynomial or standard expression, and state the final area.

1
Enough for 1 mark
The area between two curves y = f(x) and y = g(x) from x = a to x = b is ∫_a^b [upper curve - lower curve] dx, provided the same curve stays above on the whole interval.
2
Add for 2 marks
Example: The area between y = x and y = x² from x = 0 to x = 1 is ∫₀¹ (x - x²) dx = [x²/2 - x³/3]₀¹ = 1/2 - 1/3 = 1/6 square unit.
3
Add for 3 marks
This concept applies when a closed region is formed by two curves or by curves along with given boundaries.

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