Question

How do you evaluate the definite integral by the limit definition given :
$\int {6dx}$ from ${\text{ }}\left[ {4,10} \right]$

Answer 1

Hint : Definite integrals are those type of integrals that are given by boundary conditions unlike the indefinite integral which are not accompanied by boundary conditions . For example boundary conditions here are $\left[ {4,10} \right]$ . We will have to integrate the given integral by first integrating the given integral using the standard formula like we do for indefinite integrals : $\int {a{x^n}} {\text{ }}dx = \dfrac{{a{x^{n + 1}}}}{{n + 1}}$ +c but we then have to apply the limits conditions on the answer we can do that by putting values of boundary conditions on the answer, first we put higher boundary value then lower boundary value and then subtract answer calculate from higher value from the answer calculated from the lower values.

Complete step-by-step answer :
The given integral is $\int {6dx}$
We can solve the integral using the standard formula
$\int {a{x^n}} {\text{ }}dx = \dfrac{{a{x^{n + 1}}}}{{n + 1}} + c$
Comparing $\int {a{x^n}} {\text{ }}dx$ with $\int {6dx}$ , we get $a = 6$ and $n = 0$ (since we know ${\text{ }}{x^0} = 1$ )
Upon integrating we get,
$\int {6dx}$ $=$ ${\text{6}}x + c$ , where c is the constant of integration
Now we apply the boundary conditions to the solution obtained which are $\left[ {4,10} \right]$
$\left( {6x + c} \right)_4^{10}$ , Now we replace inside $x$ the bracket with both higher and lower limits and then subtract the higher limit from lower limit.
$\left( {6x + c} \right)_4^{10} = \left( {6*10 + c} \right) - \left( {6*4 + c} \right)$ ,
Upon solving we get,
$\left( {6x + c} \right)_4^{10} = {\text{ (60 + c)}} - (24 + c)$
$\left( {6x + c} \right)_4^{10} = {\text{ 36}}$
Hence we see the solution is ${\text{36}}$ .
So, the correct answer is “36”.

Note : Always remember that there does not remain a constant of integration ${\text{k}}$ or $c$ when we integrate under boundary conditions ie definite integrals unlike indefinite integrals where constant of integration is to be added after we evaluate the given indefinite integral.