Lehigh University Math 21 - The Derivative

Area between curves

The area between curves y = f(x) and y = g(x) , for, say x = a up to x = b , as pictured,

is found in the following process: Break the domain [a,b] into a bunch of subintervals, each of width Dx . Over each little piece, say centered at the point x_i , the sliver of area between the curves is a thin almost rectangular region, of area

DA_i = (g(x_i)-f(x_i))Dx,

approximately.

The total area, then, is

n
å
i = 1

(g(x_i)-f(x_i))Dx =

n
å
i = 1

DA_i,

at least approximately. Of course, as the width Dx of the slices gets smaller, the approximation to the true area improves, so that the exact area A is:

A =
lim
n® ¥
n
å
i = 1
(f(x_i)-g(x_i))Dx,

which works out to be the same limit as the integral:

ó
õ b

a
(f(x)-g(x))dx,

so this integral measures the area between the curves. Note that, as with the picture, the endpoints x = a and x = b may or may not be places where the curves intersect.

Note that here the integral is an exact value for the area, which is approximated by the sums leading up to the integral. That's the general idea with applications of integration; you approximate a situation with geometrically reasonable models, and as the approximation gets finer and finer, two things happen:

The approximations approach the real object to be measured, and
The formula becomes the integral, the limit of Riemann sums, of some function.

Example 1 Find the areas enclosed by the curves y = x² and y = 2x+1 .

Exercise 1 Find the areas enclosed by the curves y = x and y = x³ (this one has a bit of a catch).

Area =

Example 2 Find the areas enclosed by the curves y = arctan(x) and y = px/4 . The constants are supposed to help.

Exercise 2 Find the areas enclosed by the curves

y =
Ö

4-x²

, y = x, and y = -x.

Area =

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