Integration Question 1a

1. Find the Riemann sums for the following functions.

a) from x 1 to x 5 and n 4.

1.Find the Riemann sums for the following functions.

a) from x 1 to x 5 and n 4.

Answer:

a) -8

1.Find the Riemann sums for the following functions.

a) from x 1 to x 5 and n 4.

Solution:

a) Find the Riemann sum of from x 1 to x 5 for n 4.

hence

i      

Substituting in the Riemann sum,

Definition: For a positive function f(x) there may be an area under the curve. If such an area exists between the values of x

a and x

b, where a \textless b, it is called the definite integral of the function f(x) between the limits of a and b and may be expressed as

If the area under f(x) is divided into n narrow strips between a and b with equal width of

, as shown above, then;

For constant Where,

, n

the number of rectangles and f(xi) is the value of f(x) at

. Such a sum is called a Riemann sum. If the limit exists, the area under the curve is exactly equal to the limit of the sum as

This is called the Riemann integral of the function f(x) and expressed as;

Note that as

Hence the Riemann integral of the function f(x) may be written as: