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  • Determine the area of a region between two curves by integrating with respect to the independent variable.
  • Find the area of a compound region.
  • Determine the area of a region between two curves by integrating with respect to the dependent variable.

In Introduction to Integration , we developed the concept of the definite integral to calculate the area below a curve on a given interval. In this section, we expand that idea to calculate the area of more complex regions. We start by finding the area between two curves that are functions of x , beginning with the simple case in which one function value is always greater than the other. We then look at cases when the graphs of the functions cross. Last, we consider how to calculate the area between two curves that are functions of y .

Area of a region between two curves

Let f ( x ) and g ( x ) be continuous functions over an interval [ a , b ] such that f ( x ) g ( x ) on [ a , b ] . We want to find the area between the graphs of the functions, as shown in the following figure.

This figure is a graph in the first quadrant. There are two curves on the graph. The higher curve is labeled “f(x)” and the lower curve is labeled “g(x)”. There are two boundaries on the x-axis labeled a and b. There is shaded area between the two curves bounded by lines at x=a and x=b.
The area between the graphs of two functions, f ( x ) and g ( x ) , on the interval [ a , b ] .

As we did before, we are going to partition the interval on the x -axis and approximate the area between the graphs of the functions with rectangles. So, for i = 0 , 1 , 2 ,… , n , let P = { x i } be a regular partition of [ a , b ] . Then, for i = 1 , 2 ,… , n , choose a point x i * [ x i 1 , x i ] , and on each interval [ x i 1 , x i ] construct a rectangle that extends vertically from g ( x i * ) to f ( x i * ) . [link] (a) shows the rectangles when x i * is selected to be the left endpoint of the interval and n = 10 . [link] (b) shows a representative rectangle in detail.

Use this calculator to learn more about the areas between two curves.

This figure has three graphs. The first graph has two curves, one over the other. In between the curves is a rectangle. The top of the rectangle is on the upper curve labeled “f(x*)” and the bottom of the rectangle is on the lower curve and labeled “g(x*)”. The second graph, labeled “(a)”, has two curves on the graph. The higher curve is labeled “f(x)” and the lower curve is labeled “g(x)”. There are two boundaries on the x-axis labeled a and b. There is shaded area between the two curves bounded by lines at x=a and x=b. The third graph, labeled “(b)” has two curves one over the other. The first curve is labeled “f(x*)” and the lower curve is labeled “g(x*)”. There is a shaded rectangle between the two. The width of the rectangle is labeled as “delta x”.
(a)We can approximate the area between the graphs of two functions, f ( x ) and g ( x ) , with rectangles. (b) The area of a typical rectangle goes from one curve to the other.

The height of each individual rectangle is f ( x i * ) g ( x i * ) and the width of each rectangle is Δ x . Adding the areas of all the rectangles, we see that the area between the curves is approximated by

A i = 1 n [ f ( x i * ) g ( x i * ) ] Δ x .

This is a Riemann sum, so we take the limit as n and we get

A = lim n i = 1 n [ f ( x i * ) g ( x i * ) ] Δ x = a b [ f ( x ) g ( x ) ] d x .

These findings are summarized in the following theorem.

Finding the area between two curves

Let f ( x ) and g ( x ) be continuous functions such that f ( x ) g ( x ) over an interval [ a , b ] . Let R denote the region bounded above by the graph of f ( x ) , below by the graph of g ( x ) , and on the left and right by the lines x = a and x = b , respectively. Then, the area of R is given by

A = a b [ f ( x ) g ( x ) ] d x .

We apply this theorem in the following example.

Finding the area of a region between two curves 1

If R is the region bounded above by the graph of the function f ( x ) = x + 4 and below by the graph of the function g ( x ) = 3 x 2 over the interval [ 1 , 4 ] , find the area of region R .

The region is depicted in the following figure.

This figure is has two linear graphs in the first quadrant. They are the functions f(x) = x+4 and g(x)= 3-x/2. In between these lines is a shaded region, bounded above by f(x) and below by g(x). The shaded area is between x=1 and x=4.
A region between two curves is shown where one curve is always greater than the other.

We have

A = a b [ f ( x ) g ( x ) ] d x = 1 4 [ ( x + 4 ) ( 3 x 2 ) ] d x = 1 4 [ 3 x 2 + 1 ] d x = [ 3 x 2 4 + x ] | 1 4 = ( 16 7 4 ) = 57 4 .

The area of the region is 57 4 units 2 .

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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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