Key takeaways
- Area between curves: A = ∫ₐᵇ [top − bottom] dx.
- Find where the curves meet by solving f(x) = g(x).
- Test a middle x-value to see which curve is on top.
- If the curves cross inside the interval, split the integral there.
In this guide
What is the area between two curves?
Draw two curves on the same graph. Between two x-values, one curve is on top and the other is below. The region in between is like a strip of land between two winding roads. Its area tells you how much space the gap covers.
The idea: slice the region into thin vertical strips. Each strip's height is top − bottom, and its width is a tiny dx. Add up all the strips with an integral. It's the same as finding the area under the top curve and taking away the area under the bottom curve.
Why does the area between curves matter?
Most real shapes aren't bounded by the x-axis. A lens, a leaf, a crescent moon or a curved road section is bounded by two curves. The 'top minus bottom' idea handles all of them with one simple integral.
It also measures gaps between two quantities over time. If one curve is income and another is spending, the area between them is the total saved.
Between x = 0 and x = 1, the curves y = √x and y = x² enclose a leaf shape with area exactly 1/3, a third of the whole unit square.
The formula
| Symbol | What it means | Example |
|---|---|---|
| A | Area between the curves, in square units | 1/3 ≈ 0.3333 |
| f(x) | The upper curve | √x = x^0.5 |
| g(x) | The lower curve | x² |
| a, b | Left and right edges of the region (in the general formula) | 0 and 1 |
In this calculator the curves are f(x) = xᵃ and g(x) = xᵇ, and the interval is fixed at 0 to 1. Here a and b are the powers, not the limits. Between 0 and 1, a smaller power gives a higher curve: at x = 0.25, √x = 0.5 but x² = 0.0625. Enter the smaller power as a. The calculator uses the exact result 1/(a + 1) − 1/(b + 1) and gives its absolute value, so the answer is never negative.
How to find the area between curves step by step
- 1Find where the curves meet. Solve f(x) = g(x). Those x-values are your limits.
- 2Decide which curve is on top. Plug a middle x-value into both.
- 3Set up top minus bottom. Write
∫ₐᵇ [top − bottom] dx. - 4Integrate each part. With the power rule, xᵏ becomes
xᵏ⁺¹ ÷ (k + 1). - 5Evaluate and subtract. Put in b, then a, and subtract.
Now try your own pair of powers. The calculator below shows every step of the integration.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Subtracting in the wrong order. Bottom − top gives a negative number. Area is always positive.
- Guessing the limits. Solve f(x) = g(x) to find where the curves really meet.
- Mixing up the calculator's a and b. Here they are powers, and the interval is always 0 to 1.
- Forgetting square units. Area is in units², like m² or cm².
Tips and tricks
- Estimate: the area can't be bigger than (width) × (largest gap). For √x and x² that is
1 × 0.47, so 1/3 is sensible. - Sideways regions: if the region is easier to slice horizontally, use right minus left and integrate in y.
- Spin it: rotate the region around an axis to get a volume with the disk method calculator.
Where you'll use it in real life
- Economics: consumer and producer surplus, the area between a demand curve and a price line.
- Engineering: the area of odd shapes, like a lens or a leaf-shaped metal plate.
- Design: the area of curved logos and shapes in graphics software.
- Data: the total gap between two trends over time, such as income and spending.
Quick summary
To find the area between two curves, find where they meet, work out which is on top, and integrate top minus bottom. Split the integral wherever they cross. Use the calculator above for curves of the form xᵃ and xᵇ on [0, 1], with every step shown.
Worked example
A designer draws a leaf-shaped logo trapped between the curves y = √x (on top) and y = x² (below), which meet at x = 0 and x = 1. How much area does the leaf cover? Since √x = x^0.5, enter a = 0.5 and b = 2.
- 01Area between f(x) = x^0.5 and g(x) = x^2 on [0, 1]
- 02A = ∫[0,1] (x^0.5 − x^2) dx
- 03= [x^1.5/1.5 − x^3/3]₀¹
- 04= (1/1.5) − (1/3)
- 05= 0.6667 − 0.3333
- 06= 0.3333
These numbers are pre-filled in the calculator above.
Frequently asked questions
How do I know which curve is on top?
Pick any x-value between the limits and plug it into both functions. The one with the bigger value is on top there. For example, at x = 0.5, x = 0.5 is larger than x² = 0.25.
What if the curves cross in the middle?
Split the interval at each crossing point. Integrate (top − bottom) on each piece, making sure you use the right top curve each time, then add the areas.
What is the area between y = x and y = x² from 0 to 1?
It is 1/2 − 1/3 = 1/6, which is about 0.1667 square units.
Can the area between curves be negative?
No. If you get a negative number, you subtracted bottom minus top. Swap the order or take the absolute value, as this calculator does.
