Areas between curves (WACE Year 12 Mathematics Specialist Unit 4)

What this dot point is asking

SCSA wants you to set up and evaluate the area between two curves, locate the boundaries by solving for intersections, and handle regions where the upper and lower curves change.

Top minus bottom

The integrand is the vertical gap between the curves at each $x$. Because we subtract the lower curve, the result is positive and there is no need to worry about whether either curve dips below the $x$-axis: the difference handles that automatically.

Finding the limits

The limits $a$ and $b$ are usually the $x$-coordinates of the intersection points, found by solving $f(x) = g(x)$. Sketch the region to confirm which curve is on top, since that determines the order of subtraction.

Why top minus bottom always works

The reason the formula needs no absolute-value signs is geometric. At a fixed $x$, the height of the thin strip between the curves is the upper $y$-value minus the lower $y$-value, which is $f(x) - g(x)$ whenever $f$ is on top. Summing these strip areas as the width $dx \to 0$ is exactly the definite integral. Because $f(x) - g(x) \ge 0$ throughout the interval, the integral is positive automatically, even where both curves are below the $x$-axis, because shifting both curves up by the same amount leaves their difference unchanged. This is why you should never integrate $f$ and $g$ separately and subtract the signed areas: subtract the functions first, then integrate once.

When the curves cross inside

Integrating with respect to y

When the region is bounded more naturally on the left and right, it can be cleaner to write $x$ as a function of $y$ and integrate $\int_c^d [x_{\text{right}} - x_{\text{left}}]\,dy$. Choose the variable that makes the bounds simplest. A region trapped between a sideways parabola $x = y^2$ and a line, for instance, has a single clean description in terms of $y$ but would need splitting in terms of $x$, so the choice of variable can turn two integrals into one.

Choosing the right strip orientation

Deciding whether to integrate in $x$ or in $y$ is a judgement worth making before any algebra. Use vertical strips (integrate in $x$) when the region has a clear top curve and bottom curve over a single $x$-interval. Use horizontal strips (integrate in $y$) when the region has a clear right boundary and left boundary over a single $y$-interval, which is typical when the curves are given as $x$ in terms of $y$ or open sideways. If neither orientation avoids a split, pick whichever produces the simpler integrand. A quick sketch with one sample strip drawn in shows immediately which curve is the upper or right boundary and which is the lower or left boundary.

A note on signed area versus enclosed area

The phrase area between curves always means a positive, enclosed area, so the top-minus-bottom rule with a split at every crossing is the safe method. This is different from a single definite integral $\int_a^b f(x)\,dx$, which is a signed area that can be negative where $f$ lies below the axis. SCSA questions are explicit about which they want: an enclosed area is always positive, whereas a net signed value may be requested in a motion or accumulation context. Reading the command word carefully decides whether to split for positivity or to leave the integral signed.