# LAB 4 - THE DEFINITE INTEGRAL

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```					                 PROJECT 6 - THE DEFINITE INTEGRAL
paul vaz

1. Fill in the Table given below with the appropriate left- and right-hand sums with n
2

e
 x2
subdivisions for the integral                       dx . Use the table to estimate the value of the
1
definite integral.

n                      Leftsum                Rightsum
5
25
500

2. (a) Find the total area between the curve y  sin( x 2 ) and the horizontal axis from
x = 0 to x  2 .

2

 sin( x
2
(b) Compute                    ) dx .
0

3. Assume the population, P, of Mexico(in millions) between 1980 and 1990 is given by
P(t )  67 .38 (1.026 ) t ,
where t is the number of years since 1980.

What was the average population of Mexico between 1980 and 1990?

4. The rate at which the world's oil is being used is continuously increasing. Suppose the
rate ( in billions of barrels per year) is given by the function r  f (t ) , where t is
measured in years and t = 0 is the start of 1990.

(a) Write a definite integral which represents the total quantity of oil used between the
start of 1990 and the start of 1995.

(b) Suppose r  32e 0.05t . Using a left-hand sum with five subdivisions, find an
approximate value for the total quantity of oil used between the start of 1990 and the
start of 1995.

(c) Interpret the first two terms in the sum from part (b) in terms of oil consumption.
Useful Commands:
with(student):
 leftbox(f(x), x=a..b, n, shading=color);
 leftsum(f(x), x=a..b, n);
 value(%);
 evalf(%);
_____________________________________________________________________
with(student):
 rightbox(f(x), x=a..b, n, shading=color);
 rightsum(f(x), x=a..b, n);
 value(%);
> evalf(%);
____________________________________________________________________
f:=expr;
 plot( f,x=a..b);
 fsolve(f=0,x=p..q);
 int(f,x=a..c);
____________________________________________________________________

 k* int(g(x), x=a..c);

```
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