APPLICATIONS OF INTEGRAL CALCULUS OF AREAS
1. THE AREA WHEN A POSITIVE FUNCTION IS USED

Given a continuous function f which is positive on the interval [a,b], the area bounded by the graph y=f(x), the X-axis, the line x=a and the line x=b, can be expressed as area.gif (1200 bytes)
1.- Find the area bounded by the graph of the function problema1.gif (1057 bytes), the X-axis and the vertical lines x=-1 and x=2

Change the value of the step parameter to see the different stages before finding the solution.

2.- Find the area bounded by the function (x-2)ex and the X-axis between 2 and 3.

3.- Find the area bounded by the curve y=-x2, the X-axis and the straight lines x=2 and x=4. What has happened?

2. THE AREA WHEN A NEGATIVE FUNCTION IS USED 
4.- If the function f is negative on the interval [a,b] what is the sign of the integral of f on this interval? In that case could the area be the same as the integral?

5.- Check that the areas bounded by the X-axis, the straight lines x=a and x=b and the graphs of the functions f and -f are identical.

The P control (which can be dragged) allows us to shade in the graph of the function

6.- Check that both the lower and upper Reimann sums of the integral of f and -f are opposite. Deduce the relation that exists between the integrals on the same interval of the functions f and -f.

The red and turquoise rectangles can be moved by dragging them using the corresponding controls. They represent the under and over estimations of the area respectively.

The n parameter represents the number of subintervals of the partition and therefore determines the base of the rectangles

Given a continuous function f which is negative on the interval [a,b], the area bounded above the graph y=f(x) and by the X-axis and the straight lines x=a and x=b, can be expressed as areaneg.gif (1211 bytes)
3. THE AREA BOUNDED BY ANY FUNCTION

Given that we are working with continuous functions, we know from Bolzano's theorem that the function over the interval can be either positive or negative or that if there is an x value which is positive and another which is negative that there is a value in the middle which is equal to zero.

7.- Find the area bounded by the graph of the function f(x)=0.075(x+2)3-0.5(x+2)2-0.025x+1, the straight lines x=-3, x=5 and the X-axis. Clue: the points where the graph cuts the X-axis between -3 and 5 are x1=-0.37 and x2=4.34

Change the value of the step parameter to check the method and the solution.

8.- Find the area bounded by the graph of the cosine function and the X-axis between 0 and 2*p .

9.- What is the value of the definite integral of an odd function on an interval [-a,a]

4. AN AREA BOUNDED BY TWO FUNCTIONS

Take the two functions f and g. The objective is to find the area bounded by the graphs of both functions and the straight lines x=a and x=b.

10.- Find the area bounded by the functions f(X)=x(x+1)(x-3)/10+3 and  g(x)=0.08*(x-0.75)2+1, between a=-2 and b=4

Increase the step parameter to check the solution.

11.- Deduce a general formula which can be used to find the area bounded by two functions such that on the interval [a,b] the following is true: g(x)<f(x).

12.- What happens if one or both of the functions are not positive?

When the situation parameter is set at 2, the M control can be used to move functions f and g vertically.
13.- Show that, in general, the area bounded by two functions f and g where g<f ,is equal to the definite integral between a and b of the difference function f-g, regardless of the sign of either function.

14.- What about when the graphs of functions f and g cross?
Enrique Martínez Arcos
Spanish Ministry of Education. Year 2001

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