Mathematics

Problem

Let

f

and

g

be continuous on the closed interval

[a,c]

and differentiable on

(a,c)

. Suppose that •

f(a)=g(a)

f(b)=g(b)

, and

f(c)=g(c)

for some numbers

a<b<c

, and •

f(x)\ne g(x)

for all

x

in the open intervals

(a,b)

and

(b,c)

. Throughout this problem, interpret "the region between the graphs of

f

and

g

[a,c]

" as the finite planar region bounded only by the two graphs and the vertical lines

x=a

and

x=c

. (a) Explain why the region between the graphs of

f

and

g

[a,c]

can be decomposed into exactly two smaller regions, one lying entirely between

x=a

and

x=b

and the other lying entirely between

x=b

and

x=c

. Justify your answer without referencing any particular formulas for area. (b) Prove that the total area of the region described above is equal to

\displaystyle \int_a^{c} |f(x)-g(x)|\,dx

. Your explanation must rely on properties of definite integrals and the fact that

|u|=\pm u

according to the sign of

u

. (c) Suppose instead that

f(x)-g(x)

does not change sign on

[a,c]

. Describe how the integrand in part (b) simplifies and explain, in complete sentences, why this agrees with the usual "top minus bottom" formula many students memorize.

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