Area Between Curves

You’ll be able to

Set up an integral for the area between two curves
Determine which function is on top and find the bounds from intersections
Choose between integrating with respect to x or y

Top minus bottom, summed up

The area of the region trapped between two curves y = f(x) (upper) and y = g(x) (lower) over [a, b] is the integral of the vertical gap: ∫ₐᵇ [f(x) − g(x)] dx. Each thin vertical strip has height (top − bottom) and width dx; the integral adds up infinitely many such strips. Because you always subtract bottom from top, the integrand stays nonnegative and the area comes out positive.

Finding the bounds and the top

The limits a and b are usually the x-coordinates where the curves intersect — set f(x) = g(x) and solve. Between consecutive intersections one curve stays above the other; test a sample point to see which. If the curves switch which is on top within the region, split the integral at the crossover and integrate each piece with the correct top-minus-bottom order, adding the results.

Area between curves

A = ∫ₐᵇ [ f(x) − g(x) ] dx, with f on top and g on the bottom

Integrate top minus bottom. If integrating in y instead, it becomes right minus left: ∫ [x_right − x_left] dy.

Worked example

Find the area between y = x and y = x² from their intersections.

1.Find intersections: x = x² gives x² − x = 0, so x = 0 and x = 1 — these are the bounds.
2.Decide the top: at x = 0.5, y = x gives 0.5 and y = x² gives 0.25, so y = x is on top.
3.Set up the integral: A = ∫₀¹ (x − x²) dx.
4.Evaluate: [x²/2 − x³/3]₀¹ = (1/2 − 1/3) = 3/6 − 2/6 = 1/6.

Answer: The area is 1/6. The line y = x lies above the parabola y = x² on (0, 1), so the strip height is x − x².

Watch out

Always subtract in the order top minus bottom (or right minus left in y). Reversing it gives a negative of the true area. If you get a negative number for an area, you likely flipped the subtraction.

Checkpoint

The region between y = 4 − x² (top) and y = 0 (the x-axis) is bounded by their intersections. What integral gives the area?

Checkpoint

To find the area between two curves that intersect at x = 1 and x = 3, with f above g throughout, you integrate:

On the exam

When the region is easier to slice horizontally — for instance bounded by curves given as x = h(y) — integrate with respect to y using right minus left. Choosing the orientation that avoids splitting the region saves time on free response.

Answer the 2 checkpoints as you read.