MATH 131 -- Calculus for Physical and Life Sciences 1

Some Integration Examples

December 3, 2004

1.  Left- and Right-hand sums for   on [-1,1]:

>	with(Student[Calculus1]):

>	RiemannSum(exp(-x),x=-1..1,partition=4,method=left);

>

evalf(%);

>	RiemannSum(exp(-x),x=-1..1,partition=4,method=right);

>

evalf(%);

>	evalf(int(exp(-x),x=-1..1));

>	RiemannSum(exp(-x),x=-1..1,output=plot,partition=4,method=left,boxoptions=[color=blue],title=" ",font=[TIMES,ROMAN,12],showpoints=false,scaling=constrained);

>	RiemannSum(exp(-x),x=-1..1,output=plot,partition=4,method=right,boxoptions=[color=blue],title=" ",font=[TIMES,ROMAN,12],showpoints=false,scaling=constrained);

The following command shows an animation of the Left-hand sums as the number of

intervals in the partitions is increased.

>	RiemannSum(exp(-x),x=-1..1,output=animation,iterations=7,method=right,boxoptions=[color=blue],title=" ",font=[TIMES,ROMAN,12],showpoints=false,scaling=constrained);

2.  Approximations to

>	RiemannSum(sin(x),x=0..2*Pi,output=animation,iterations=7,method=left,boxoptions=[color=red],title=" ",font=[TIMES,ROMAN,12],showpoints=false,scaling=constrained);

3.  Area between curves:

>	RiemannSum(3*x-x^2,x=0..3,output=animation,iterations=7,method=right,boxoptions=[color=blue],title=" ",font=[TIMES,ROMAN,12],showpoints=false,scaling=constrained);

4.  Total Change of a function:

>	RiemannSum(4500*exp(-0.05*t)/(1+9*exp(-0.05*t))^2,t=0..15,partition=1000);

5.  Average value of a function:

>	RiemannSum(sin(x),x=0..Pi,output=animation,iterations=7,method=right,boxoptions=[color=blue],title=" ",font=[TIMES,ROMAN,12],showpoints=false,scaling=constrained);

>	evalf(int(sin(x),x=0..Pi));

>	evalf(2/Pi);

>	FunctionAverage(sin(x),x=0..Pi,output=plot,titlefont=[TIMES,ROMAN,12],font=[TIMES,ROMAN,12],scaling=constrained);

>

>