Area between two curves

Find the area enclosed between two graphs - a common Paper 2 integral.

In short: The area between two curves is found by integrating the difference between the upper and lower functions between their intersection points. First find the intersection x-values with the GDC, then evaluate the definite integral of the top function minus the bottom one. Sketch the region to be sure which curve is on top.

When you'd use this

Finding the area enclosed between two graphs rather than between a graph and the x-axis.
The curves' intersection points aren't nice round numbers, so you need the GDC to find them first.
One or both functions can't be integrated by hand.
Checking a by-hand "integrate the difference" calculation, including the limits.

TI-84 Plus CE Casio fx-CG50 and fx-CG100 TI-Nspire CX

At a glance: TI-84 Plus CE, Casio fx-CG50 and TI-Nspire CX compared

	TI-84 Plus CE	Casio fx-CG50 and fx-CG100	TI-Nspire CX
Key sequence	Integrate top minus bottom: MATH → 9:fnInt(Y1 − Y2, X, a, b).	Run-Matrix → ∫dx of (upper − lower) between the intersection x-values.	Use the ∫ template: ∫(f(x) − g(x)) dx from a to b (upper curve minus lower).

On a TI-84 Plus CE

Find where the curves meet (intersection) - these x-values are the limits.
Integrate top minus bottom: MATH → 9:fnInt(Y1 − Y2, X, a, b).
Integrate (upper − lower) so the area is positive.

Tip: Find the intersection points first - they are the limits of the integral.

On a Casio fx-CG50 and fx-CG100

Find where the curves meet (intersection) - these x-values are the limits.
Run-Matrix → ∫dx of (upper − lower) between the intersection x-values.
Integrate (upper − lower) so the area is positive.

Tip: Find the intersection points first - they are the limits of the integral.

On a TI-Nspire CX

Find where the curves meet (intersection) - these x-values are the limits.
Use the ∫ template: ∫(f(x) − g(x)) dx from a to b (upper curve minus lower).
Integrate (upper − lower) so the area is positive.

Tip: Find the intersection points first - they are the limits of the integral.

Related guides

Try it yourself

Here's a real IB-style question that uses exactly this technique.

Hard Calculator Paper 2 [4 marks]

Find the area enclosed between y = x² and y = x + 2.

4.5

Mark it

Correct 4 / 4 marks

Worked solution & mark scheme:

M1 Find the intersections: x = −1 and x = 2

M1 ∫ from −1 to 2 of [(x + 2) − x²] dx

A1 4.5

Common questions

When would I need to find the area between two curves in IB Maths?

Find the area enclosed between two graphs - a common Paper 2 integral. Finding the area enclosed between two graphs rather than between a graph and the x-axis. The curves' intersection points aren't nice round numbers, so you need the GDC to find them first.

How do I find the area between two curves on a TI-84 Plus CE?

1. Find where the curves meet (intersection) - these x-values are the limits. 2. Integrate top minus bottom: MATH → 9:fnInt(Y1 − Y2, X, a, b). 3. Integrate (upper − lower) so the area is positive.

How do I find the area between two curves on a Casio fx-CG50 and fx-CG100?

1. Find where the curves meet (intersection) - these x-values are the limits. 2. Run-Matrix → ∫dx of (upper − lower) between the intersection x-values. 3. Integrate (upper − lower) so the area is positive.

How do I find the area between two curves on a TI-Nspire CX?

1. Find where the curves meet (intersection) - these x-values are the limits. 2. Use the ∫ template: ∫(f(x) − g(x)) dx from a to b (upper curve minus lower). 3. Integrate (upper − lower) so the area is positive.

What should I watch out for when I find the area between two curves?

Find the intersection points first - they are the limits of the integral.

How are marks awarded when I find the area between two curves in an IB exam?

In the worked example on this page (4 marks, Paper 2), the marks are: M1: Find the intersections: x = −1 and x = 2; M1: ∫ from −1 to 2 of [(x + 2) − x²] dx; A1: 4.5.

Practise with your calculator

Questions that need this technique link back to this guide. Try one in practice mode, or see every GDC guide.