TMUA Practice Question — Worked Solution

P2·Integration·6.5Mock 0 · P2 · Q16

area between curvesmodulusproof errorcombining integralssign of an integral

Let

f

and

g

be polynomials, let

a < b

, and let

F

be any function with

F^{'} (x) = f (x) - g (x)

.A student gives the following argument for this claim:The total area of the region or regions enclosed by the curves $y = f (x)$ and $y = g (x)$ and the lines $x = a$ and $x = b$ is $∣ F (b) - F (a) ∣$ .

(I)

Let

h (x) = f (x) - g (x)

. Split the interval

a \leq x \leq b

at every point where the two curves cross. On each of the resulting intervals one curve is never below the other, so

h (x)

does not change sign there.

(II)

On any one of these intervals,

p \leq x \leq q

, the area between the curves is

\int_{p}^{q} h (x) d x

h (x) \geq 0

there, and

- \int_{p}^{q} h (x) d x

h (x) \leq 0

there.

(III)

So in either case, the area between the curves for

p \leq x \leq q

\int_{p}^{q} h (x) d x

(IV)

The intervals are contiguous and together make up

a \leq x \leq b

, so, combining the integrals over these ranges, the total area is

\int_{a}^{b} h (x) d x

(V)

Since

F^{'} (x) = h (x)

, we have

\int_{a}^{b} h (x) d x = F (b) - F (a)

whichever such

F

is used, so the total area is

∣ F (b) - F (a) ∣

Which one of the following is true?

All questions

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