Area between a curve and the x-axis

In 5 of 20 recent exams
Paper 1 or 2
3 to 7 marks

How it is set in the exam

Paper 1 (no calculator) (exact, split at the roots) and Paper 2 (calculator) (GDC), 3–7 marks. Typical (5 of 20 exam versions); the same method gives the distance travelled in kinematics, which is far more common.

Spot it in the exam

Exam phrases to spot (5)

“find the area of the region enclosed by the graph of $f$ , the $x$ -axis and the lines $x = 1$ and $x = 4$ ”
“find the area of the region enclosed by the graph of $f$ and the coordinate axes”
“write down an expression for the area of $R$ ”
“the shaded region $R$ is bounded by the curve and the $x$ -axis”
“find the total area of the shaded regions”

In disguise (2)

no lines $x = \ldots$ are given: the curve cuts the region off between two roots, so the limits are the roots
"show that $\int_0^6 f(x)\,dx = 0$ ", followed by an area: the integral is $0$ because a piece below the axis cancels a piece above, and the area is not $0$

Variants (1)

"the area of $R$ is $12$ ; find $k$ ", with $k$ in a limit or in the function: the area, written in terms of $k$ , is an equation

Looks like it, but isn't (3)

two curves, or a curve and a line other than the $x$ -axis: Area between two curves
an area measured against the $y$ -axis ( $\int \ldots\,dy$ ): HL (AHL 5.17)
an integral is asked for, even with the region drawn: Integrate: choose the method, with no split

Worked example

Part of the region is below the axis: split at the root

Paper 1 (no calculator)

Question

Let $f(x) = 2\cos x + 1$ , for $0 \le x \le \pi$ .
(a) Find the $x$ -coordinate of the point where the graph of $f$ crosses the $x$ -axis. [3]
(b) Find $\int_0^{\pi} f(x)\,dx$ . [3]
(c) Find the total area of the two regions enclosed by the graph of $f$ , the coordinate axes and the line $x = \pi$ . [5]

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Steps

Roots:
solve $f(x) = 0$ on the interval (: zeros and intersections).
Sketch
the curve between the limits, and mark where it is above and below the axis.
Write the expression
for the area before calculating. : one integral for each piece between consecutive roots, with a minus sign (or bars) on each piece below the axis. : $\int_a^b \lvert f(x) \rvert\,dx$ (definite integral).
Evaluate
and add the sizes. The area is positive.
Area given, $k$ unknown:
write the area in terms of $k$ (split at the roots first), set it equal to the value, solve, and reject the values the conditions rule out.

Common mistakes

One integral across a root,
so the pieces cancel: split at every root in the interval, or use $\lvert f \rvert$ on .
A negative area
a region below the axis has a negative integral, and its area is the size of that integral.
Integrating before finding the roots,
so the split point is missed.

More mistakes (1)

A GDC value with no integral written down
the expression earns the method mark.

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