Area Between Curves

If f is continuous and nonnegative on [a,b], then the definite integral for the area A under f(x) over the interval [a,b] is obtained in four steps,

Divide the interval [a,b] into subintervals, and use those subintervals to divide area under the curve y=f(x) into n steps
Assuming that the width of the k^th strip is , approximate the area of that strip by the area of the rectangle of width and height where is any point in the k^th subinterval
Add the approximate areas of the strips to approximate the entire area A by the Riemann sum:
Take the limit of the Riemann sum as the number of sub intervals increases and their widths approach zero. This causes the error in the approximation to approach zero and produces the following definite integral for the exact area A:

Suppose that f and g are continuous functions on an interval [a,] and f(x) >= g(x) for a <= x <= b

This means that the curve y=f(x) lies above the curve y=g(x) and that the two can touch each other but not cross. Find the area A of the region bounded above by y=f(x) , below by y=g(x) and on the sides by the lines x=a and x=b

Area Between Curves f(x) and g(x) between x=a and x=b

To solve this problem we divide the interval [a,b] into n subintervals, which has the effect of subdividing the region into n strips (second Figure above). If we assume that the width of the kth strip is delta x , then the area of the strip can be approximated by the area of the rectangle of width delta x and height function of x and y at kth interval where src="function of x at kth interval.gif"> is any point in the kth subinterval. Adding these approximations yields the following Riemann sum that approximates the area A: entire area under curve by the riemann sum

Taking the limit as n increases and the width of the subintervals approach zero yields the following definite integral for the area A between the curves:

definite integral for the exact area between curves

From the above, we have the following result:

AREA FORMULA :

If f and g continuous functions on the interval [a,b], and if f(x)>=g(x) for all x in [a,b] , then the area of the region bounded above by f(x) , below by g(x) on the left by line x=a and on the right by line x=b is

area between curves definite integral

EXAMPLE

Find the area of the region bounded above by y=x+6 and bounded below by y=x² and bounded on the sides by lines x=0 and x=3

Find Area f(x)=x+6 g(x)=x^2 Find Area f(x)=x+6 g(x)=x^2 n strips

Solution:

The region and a cross sections are shown in the figures below. The cross section extends from g(x)=x² on the bottom to f(x)=x+6 on the top. The cross section is moved through the region, then its left most position will be x=0 and its rightmost position will be x=3, Thus

area y=x+6 y=x^2 a=0 b=3.gif

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