BC 1- 2
Riemann Sums
Name:
Definition of an Integral
We've seen that we can estimate distance traveled by looking at the area under a speed function. We can also estimate the total amount of water that has leaked out of a tank by looking at the area under the function that represents the rate at which the water is leaking. More generally, we can estimate total change in a quantity by estimating the signed area bounded by the rate function over the interval in question.
At the same time, we've seen that as the step size grows smaller, the approximation given by rectangles grows better. We need to give a definition and make a few clarifications.
Definition: Suppose ƒ is continuous on a ≤ x ≤ b . A partition of the interval [ a , b ] is formed with x
0
= a , x
1
, x
2
, … x n
= b . The definite integral of ƒ from a to b , is given by if this limit exists, where c i
 x
 max
 x i
 a b f ( x ) dx

[ x i

1
, x i if n
 
, that will not guarantee that
]
 lim n

 x

0 n  i

1 f ( c i
)
 x i
, the i th
subinterval of [ a , b ].
is called the "mesh size" of the partition. As
 x

0 , then n
 
, but
 x

0 . The value of this integral is equal to the signed area bounded by the graph of ƒ, x = a , x = b , and the x -axis. However, as seen previously, the value of the integral may represent many things other than area. We'll study more applications later.
To work with this limit, it is necessary to approach the problem with patterns. Here, we'll choose all
 x to have the same length, and we'll take c i
to be the right-hand endpoint in each interval. So, in general, for the interval [ a , b ], we have
 x = and c
2
M c
1
 a
  x
 c i

(1) Express

1
4

5 x


as a limit. (Do not try to evaluate!)
BC 1-2 S12

(2) Express

2
5

4 x

1
 dx as a limit. Evaluate the limit to find the value of the integral.
(3) Express


1
4

3
 x
2
 dx as a limit. Evaluate the limit to find the value of the integral.
BC 1-2 S12