Regions with Several Crossings

The rule top minus bottom needs one curve on top for the whole interval. When two curves cross inside the interval, they trade places at each crossing, and the area has to be found in pieces.

The integral of

f - g

without the absolute value is a signed total. Pieces where

g

is on top count as negative, so it's smaller than the area whenever

g

is on top somewhere.

A sum of integrals

Example 1

Let

f(x) = x(x + 1)(x - 2)

and

g(x) = x^2 + x

. Find the total area of the regions enclosed by the two graphs.

Since

x^2 + x = x(x + 1)

\begin{aligned} &f(x) - g(x) \\ &\quad = x(x + 1)\big((x - 2) - 1\big) \\ &\quad = x(x + 1)(x - 3). \end{aligned}

The graphs cross at

x = -1

0

, and

3

. Each factor changes sign at its own zero, so the difference changes sign at each crossing. At

x = -\frac{1}{2}

it's

\frac{7}{8} > 0

, so

f

is on top on

(-1, 0)

. At

x = 1

it's

-4

, so

g

is on top on

(0, 3)

f is on top from −1 to 0 (blue), and g is on top from 0 to 3 (amber).

Expand

f - g = x^3 - 2x^2 - 3x

, with antiderivative

H(x) = \frac{x^4}{4} - \frac{2x^3}{3} - \frac{3x^2}{2}

. Its values at

-1

0

, and

3

are

-\frac{7}{12}

0

, and

-\frac{45}{4}

\begin{aligned} A &= \int_{-1}^{0} \big(f(x) - g(x)\big)\,dx \\ &\qquad + \int_{0}^{3} \big(g(x) - f(x)\big)\,dx \\[4pt] &= \left(0 + \tfrac{7}{12}\right) + \left(0 + \tfrac{45}{4}\right) \\[4pt] &= \frac{7}{12} + \frac{135}{12} = \frac{71}{6}. \end{aligned}

Integrating

f - g

from

-1

3

in one step gives

\frac{7}{12} - \frac{45}{4} = -\frac{32}{3}

, the signed total.

One integral of the absolute value

With a calculator, the absolute-value form needs no sign analysis. Find the crossings, then integrate

|f - g|

from the first to the last.

Example 2

A calculator is allowed. Find the total area of the regions enclosed by the graphs of

y = x^3 - 3x

and

y = 2\sin x + 1

, to three decimal places.

Solving

x^3 - 3x = 2\sin x + 1

with a calculator gives three solutions,

a \approx -1.868

b \approx -0.202

, and

c \approx 2.079

. Store them.

The cubic is on top from a to b (blue), and the sine curve is on top from b to c (amber).

\begin{aligned} A &= \int_{a}^{c} \left|x^3 - 3x - 2\sin x - 1\right|dx \\ &\approx 9.974. \end{aligned}

The two pieces have areas about

3.008

and

6.966

. The integral of the difference without the absolute value is about

-3.957

, the signed total.