Mathfolis

Area Between Curves

Unit 8 · Applications of Integration

Area Between Curves

The area between two curves is ∫[top − bottom] dx or ∫[right − left] dy. Find intersection points first — these become the limits. If the curves cross, split the integral at each crossing and add the absolute values.

Area Formulas

With respect to x (f ≥ g on [a,b])
A=∫ab[f(x)−g(x)] dxA = \int_a^b [f(x) - g(x)]\,dx
With respect to y (right − left)
A=∫cd[xright(y)−xleft(y)] dyA = \int_c^d [x_{\text{right}}(y) - x_{\text{left}}(y)]\,dy
Caution: When curves cross, the top/bottom relationship reverses — split the integral at every crossing point and take absolute values.
AP Tip: Integrate with respect to y when the region is easier to describe horizontally (bounded left and right rather than top and bottom).
Type 1

Area with Respect to x

Find intersections, determine which curve is on top, then integrate the difference over [a, b].

Example 1

Find the area of the region enclosed by y = x² and y = 2x + 3.

Example 2

Find the area between f(x) = 6x − x² and g(x) = x² − 2x.

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Type 2

Area with Respect to y

When curves are expressed as x = f(y), find y-intersections, then integrate [right − left] with respect to y.

Example 3

Find the area enclosed by x = y + 2 and x = y².

Example 4

Find the area enclosed by x = y² − 4 and x = 2 − y².

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Type 3

Curves That Cross — Multiple Intersections

Find all intersections, split the integral at each, and sum the absolute values of each sub-integral.

Example 5

Find the total area enclosed by y = x³ − 4x and y = 0 on [−2, 2].

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