Areas with integration

Using definite integrals for areas — the trap when a curve dips below the axis, and finding the area between two curves (or a curve and a line) as the integral of top minus bottom between their intersections.

Pure · Integration

Areas with integration

Why it works

A definite integral

\displaystyle\int_a^b y\,dx

measures the signed area between the curve and the

x

-axis from

a

b

. "Signed" is the word to watch: area above the axis counts as positive, area below as negative.The below-the-axis trap. If part of the curve lies below the

x

-axis, its integral is negative, and integrating straight across a region that is partly above and partly below gives you the difference of the two areas, not the total. To find the actual area, split the integral at each

x

-intercept and take the modulus of any negative piece before adding. For instance

\int_{-2}^{2}(x^2 - 4)\,dx = -\tfrac{32}{3}

, but the area enclosed between that curve and the axis is

\tfrac{32}{3}

— the integral came out negative only because the whole arc sits below the axis.

Area between two curves. This is the real Year 2 skill. The region trapped between an upper curve

y = f(x)

and a lower curve

y = g(x)

has area

\int_a^b \big(f(x) - g(x)\big)\,dx = \int_a^b (\text{top} - \text{bottom})\,dx.

Subtracting the lower curve from the upper one measures the vertical gap at each

x

, and integrating that gap sweeps out the region. The limits

a

and

b

are the

x

-coordinates where the curves meet, found by solving

f(x) = g(x)

.A neat bonus: because you subtract first, this works even when the region dips below the

x

-axis — the "top minus bottom" gap is positive throughout the region, so no splitting is needed. The same recipe handles the area between a curve and a straight line (the line is just one of the two functions).