Calculus/Area

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Area

Introduction[edit]

Finding the area between two curves, usually given by two explicit functions, is often useful in calculus.

In general the rule for finding the area between two curves is

 A = A_{top} - A_{bottom} \ or

If f(x) is the upper function and g(x) is the lower function

 A = \int_a^b[f(x)-g(x)]\, dx

This is true whether the functions are in the first quadrant or not.

Area between two curves[edit]

Suppose we are given two functions y1=f(x) and y2=g(x) and we want to find the area between them on the interval [a,b]. Also assume that f(x)≥ g(x) for all x on the interval [a,b]. Begin by partitioning the interval [a,b] into n equal subintervals each having a length of Δx=(b-a)/n. Next choose any point in each subinterval, xi*. Now we can 'create' rectangles on each interval. At the point xi*, the height of each rectangle is f(xi*)-g(xi*) and the width is Δx. Thus the area of each rectangle is [f(xi*)-g(xi*)]Δx. An approximation of the area, A, between the two curves is

 A := \sum_{i=1}^{n} [f(x_{i}^{*})-g(x_{i}^{*})]\Delta x.

Now we take the limit as n approaches infinity and get

A = \lim_{n \to \infty} \sum_{i=1}^{n} [f(x_{i}^{*})-g(x_{i}^{*})]\Delta x

which gives the exact area. Recalling the definition of the definite integral we notice that

A = \int_a^b[f(x)-g(x)]\,dx.

This formula of finding the area between two curves is sometimes known as applying integration with respect to the x-axis since the rectangles used to approximate the area have their bases lying parallel to the x-axis. It will be most useful when the two functions are of the form y1=f(x) and y2=g(x). Sometimes however, one may find it simpler to integrate with respect to the y-axis. This occurs when integrating with respect to the x-axis would result in more than one integral to be evaluated. These functions take the form x1=f(y) and x2=g(y) on the interval [c,d]. Note that [c,d] are values of y. The derivation of this case is completely identical. Similar to before, we will assume that f(y)≥ g(y) for all y on [c,d]. Now, as before we can divide the interval into n subintervals and create rectangles to approximate the area between f(y) and g(y). It may be useful to picture each rectangle having their 'width', Δy, parallel to the y-axis and 'height', f(yi*)-g(yi*) at the point yi*, parallel to the x-axis. Following from the work above we may reason that an approximation of the area, A, between the two curves is

 A := \sum_{i=1}^{n} [f(y_{i}^{*})-g(y_{i}^{*})]\Delta y.

As before, we take the limit as n approaches infinity to arrive at

A = \lim_{n \to \infty} \sum_{i=1}^{n} [f(y_{i}^{*})-g(y_{i}^{*})]\Delta y,

which is nothing more than a definite integral, so

A = \int_c^d[f(y)-g(y)]\,dy.

Regardless of the form of the functions, we basically use the same formula.

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Area