Question

How do you evaluate the definite integral by the limit definition given `int 4dx` from [0,3]?

Answer 1

`12`

Explanation:

`int_0^3 4 dx = [4x]_0^3`

`= [4*3] - [4 * 0] = 12`

Answer 2

see below

Explanation:

`int_0^3 4 dx`,

Integral by limit definition is given by

`lim_(n ->oo) sum_i^n f(c_i) delta x`

therefore,

`delta x = (3-0)/n = 3/n`
`c_i = a + delta x = 0 + 3/ni = 3/ni`

`int_0^3 4 dx= lim_(n ->oo) sum_i^n f(c_i) delta x`,

`= lim_(n ->oo) sum_i^n f(3/ni) 3/n`

for `i = 1,2,3,...,n -> f(x) =4`

`= lim_(n ->oo) sum_i^n 4 *3/n = lim_(n ->oo) sum_i^n 12/n`

`=lim_(n ->oo) n 12/n = lim_(n ->oo) 12 = 12`