# Integration Question 1b

**1. Find the Riemann sums for the following functions.**

b) from x 2 to x 8 and n 12.

**1.Find the Riemann sums for the following functions.**

b) from x 2 to x 8 and n 12.

**Answer:**

b) 51

**1.Find the Riemann sums for the following functions.**

b) from x 2 to x 8 and n 12.

**Solution:**

b) Find the Riemann sum of from x 2 to x 8 and n 12.

hence

is the sum of 12 terms of an arithmetic series with a 3 and d 1.

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: