Area between two curves

The formula

• The area of the region enclosed between two curves is:

$$A = \int_a^b \left[f(x) - g(x)\right]dx$$

• Each part:

  • $f(x)$ — the upper curve on the interval
  • $g(x)$ — the lower curve
  • $a$, $b$ — the $x$-values of the points of intersection, usually

• The subtraction must be top minus bottom. Getting it the other way round gives the right magnitude with the wrong sign.

Why the sign issue disappears

• Because you integrate the difference, the region between the curves is measured directly — and it does not matter whether either curve is above or below the $x$-axis.
• If both are below the axis, the difference is still positive as long as $f$ is the upper one.
• This is a genuine simplification, and it is why "area between curves" is often easier than "area under a curve".

The method

• Find the points of intersection by setting $f(x) = g(x)$ and solving. These give the limits.

• Sketch, or test a point between the intersections to see which curve is on top.

• Set up $\int (\text{upper} - \text{lower})$.

• Integrate and evaluate.

• Worked through — the area between $y = x^2$ and $y = 2x$:

  • Intersections: $x^2 = 2x$, so $x(x-2) = 0$, giving $x = 0$ and $x = 2$
  • Test $x = 1$: the line gives 2, the parabola gives 1 — the line is on top
  • $$A = \displaystyle\int_0^2 \left(2x - x^2\right)dx = \left[x^2 - \frac{x^3}{3}\right]_0^2 = 4 - \frac{8}{3} = \frac{4}{3}$$

When the curves cross inside the interval

• If the two curves swap over partway through, the region must be split at each crossing point.
• On each piece, subtract in the order appropriate to that piece.
• Then add the results — both will be positive if done correctly.

Limits that are not intersections

• Sometimes the region is bounded by given vertical lines rather than by the curves meeting.
• Use the given $x$-values as limits, but still check which curve is on top on that interval, and still check whether they cross inside it.

Horizontal strips

• If the region is more naturally described by $y$-limits — for example bounded left and right by curves — integrate with respect to $y$:

$$A = \int_c^d \left[x_{\text{right}} - x_{\text{left}}\right]dy$$

• Rearrange both equations to make $x$ the subject first.
• Choose whichever direction needs fewer splits.

Checking your answer

• An area between curves must be positive. A negative result means the subtraction was the wrong way round — fix the order rather than just dropping the sign.
• Estimate from the sketch. If the region looks like roughly a $2 \times 1$ rectangle and your answer is 30, something is wrong.