Area Between Curves Formula

Area between curves are the area of the region enclosed between two functions f(x) and g(x) from x = a to x = b, computed as A = _a^b |f(x) - g(x)|\,dx.

The Formula

A=∫ab[f(x)−g(x)] dxwhere f(x)≥g(x) on [a,b]A = \int_a^b [f(x) - g(x)]\,dx \quad \text{where } f(x) \geq g(x) \text{ on } [a, b]
If the curves cross, split into separate integrals at each intersection point.

When to use: To find the area between two curves, subtract the lower curve from the upper curve and integrate. It's like finding the area under the top curve and subtracting the area under the bottom curve—the difference is the area of the 'sandwich' between them.

Quick Example

Area between f(x)=x2f(x) = x^2 and g(x)=xg(x) = x from x=0x = 0 to x=1x = 1:
A=∫01(x−x2) dx=[x22−x33]01=12−13=16A = \int_0^1 (x - x^2)\,dx = \left[\frac{x^2}{2} - \frac{x^3}{3}\right]_0^1 = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}

Notation

Top minus bottom (for vertical slices) or right minus left (for horizontal slices with dydy integration).

What This Formula Means

The area of the region enclosed between two functions f(x)f(x) and g(x)g(x) from x=ax = a to x=bx = b, computed as A=∫ab∣f(x)−g(x)∣ dxA = \int_a^b |f(x) - g(x)|\,dx.

To find the area between two curves, subtract the lower curve from the upper curve and integrate. It's like finding the area under the top curve and subtracting the area under the bottom curve—the difference is the area of the 'sandwich' between them.

Formal View

If ff and gg are continuous on [a,b][a, b] with f(x)≥g(x)f(x) \geq g(x), then the area of the region between them is A=∫ab[f(x)−g(x)] dxA = \int_a^b [f(x) - g(x)]\,dx. In general: A=∫ab∣f(x)−g(x)∣ dxA = \int_a^b |f(x) - g(x)|\,dx.

Worked Examples

Example 1

easy
Find the area between f(x)=x+2f(x) = x+2 and g(x)=x2g(x) = x^2 from x=−1x=-1 to x=2x=2.

Answer

92\frac{9}{2}

First step

1
Intersections: x+2=x2⇒(x−2)(x+1)=0⇒x=−1,2x+2=x^2 \Rightarrow (x-2)(x+1)=0 \Rightarrow x=-1,2 (endpoints).

Full solution

  1. 2
    At x=0x=0: f(0)=2>g(0)=0f(0)=2 > g(0)=0, so f≥gf \geq g throughout.
  2. 3
    A=∫−12(x+2−x2) dx=[x22+2x−x33]−12A = \int_{-1}^{2}(x+2-x^2)\,dx = \left[\frac{x^2}{2}+2x-\frac{x^3}{3}\right]_{-1}^{2}.
  3. 4
    F(2)=2+4−83=103F(2) = 2+4-\frac{8}{3} = \frac{10}{3}; F(−1)=12−2+13=−76F(-1)=\frac{1}{2}-2+\frac{1}{3}=-\frac{7}{6}.
  4. 5
    A=103+76=206+76=276=92A = \frac{10}{3}+\frac{7}{6} = \frac{20}{6}+\frac{7}{6} = \frac{27}{6} = \frac{9}{2}.
Identify intersection points, confirm which function is on top, then integrate the difference over the interval.

Example 2

hard
Find the total area enclosed between y=x3−xy = x^3 - x and the xx-axis.

Example 3

medium
Find the area between y=x2y=x^2 and y=9y=9.

Common Mistakes

  • Integrating bottom minus top — always upper curve minus lower, or you get a negative area.
  • Not splitting at intersections when the curves cross — top and bottom swap, so each piece needs its own top-minus-bottom integral.
  • Forgetting to find the bounds by solving f(x)=g(x)f(x)=g(x) — the limits are usually the intersection points, not given.

Why This Formula Matters

This extends the single definite integral (area under one curve) to regions bounded by two, the setup for volumes of revolution next. The make-or-break step is determining which function is on top over each piece — if the curves cross, top and bottom swap, and you must split the integral at the crossings. Recognizing it by "Is the region bounded above and below by two curves, so I integrate upper minus lower?" — rather than by familiar numbers — is what lets a student tell it apart from area under one curve and definite integral (signed) and volumes of revolution in a mixed problem set.

Frequently Asked Questions

What is the Area Between Curves formula?

The area of the region enclosed between two functions f(x)f(x) and g(x)g(x) from x=ax = a to x=bx = b, computed as A=∫ab∣f(x)−g(x)∣ dxA = \int_a^b |f(x) - g(x)|\,dx.

How do you use the Area Between Curves formula?

To find the area between two curves, subtract the lower curve from the upper curve and integrate. It's like finding the area under the top curve and subtracting the area under the bottom curve—the difference is the area of the 'sandwich' between them.

What do the symbols mean in the Area Between Curves formula?

Top minus bottom (for vertical slices) or right minus left (for horizontal slices with dydy integration).

Why is the Area Between Curves formula important in Math?

This extends the single definite integral (area under one curve) to regions bounded by two, the setup for volumes of revolution next. The make-or-break step is determining which function is on top over each piece — if the curves cross, top and bottom swap, and you must split the integral at the crossings. Recognizing it by "Is the region bounded above and below by two curves, so I integrate upper minus lower?" — rather than by familiar numbers — is what lets a student tell it apart from area under one curve and definite integral (signed) and volumes of revolution in a mixed problem set.

What do students get wrong about Area Between Curves?

The procedure for area between curves is the easy part; the trap is integrating bottom minus top. Asking "Is the region bounded above and below by two curves, so I integrate upper minus lower?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Area Between Curves formula?

Before studying the Area Between Curves formula, you should understand: definite integral, fundamental theorem.

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How to Integrate Rational Functions: Long Division and Partial Fractions →
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