Area Under A Curve

an example

We are going to investigate the area of the region between the graph of the x-axis and the graph of  between x = -1 and x = 3.

>	with(plots):

Warning, the name changecoords has been redefined

>	with(Student[Calculus1]):

>	f:=x->x^3-2*x^2-3*x+10;

>	plot(f(x),x=-2..3.5,y=-5..20,thickness=3);

Here is a picture of the region whose area is to be calculated.

>	plot(f(x),x=-1..3,y=-3..12,thickness=3,filled=true);

First we will approximate the area using a "right" approximation with 20 subintervals.

>	ApproximateInt(f(x),x=-1..3,method=right,partition=20,output=plot,thickness=3);

The sum command can be used to sum the areas of the 20 approximating rectangles.

>	sum(f(-1+i/5)/5,i=1..20);

>

evalf(%);

The exact area can be computed using a definite integral and the Fundamental Theorem of Calculus.

>	Int(f(x),x=-1..3);

>

evalf(%);

>	int(f(x),x=-1..3);

Here is the area approximated using 200 approximating rectangles.

>	sum(f(-1+i/50)/50,i=1..200);

>

evalf(%);

>	ApproximateInt(f(x),x=-1..3,method=right,partition=200,output=plot,thickness=3);

Here is an animation going from 4 subintervals to 128 subintervals.

>	ApproximateInt(f(x),x=-1..3,method=right,partition=4,output=animation,thickness=2);

Here is an animation using a "midpoint" approximation.

>	ApproximateInt(f(x),x=-1..3,method=midpoint,partition=4,output=animation,thickness=2);

The animation below uses a "left" approximation.

>	ApproximateInt(f(x),x=-1..3,method=left,partition=4,output=animation,thickness=2);

>