For fx 2x find a formula for the Riemann sum obtained by div

For f(x) =2x, find a formula for the Riemann sum obtained by dividing the interval [2.5] subintervals and using the right hand endpoint for each c_k. Simplify the sum and take the limit as n--> infinity to calculate the area under the curve over [2,5]

please show all of your work as be as descriptive as you can I appreciate your help thank you!

Solution

a=2, b =5, x =(b-a)_/n =(5-2)_/n =3_/n

f(x)=2x

x_i=a+ ix =2+i(3_/n)

f(x_i)=2(2+3(i_/n))

the area under the curve over [2,5] is =lim_{n-> [i =1 to n]}f(x_i)x

=lim_{n-> [i =1 to n]}2(2+3(i_/n))(3_/n)

=lim_{n-> [i =1 to n]}6((2_/n)+3(i_/n²))

=lim_n-> 6((2n_/n)+3(n(n+1)_/2n²))

=lim_n-> 6(2 +3((1+(1/n))_/2))

= 6(2 +3((1+0)_/2))

=6(2 +3(1_/2))

=6(2 +(3_/2))

=6(7_/2)

=3*7

=21

the area under the curve over [2,5] is 21