Question

Evaluate the following by definition of
Reumann's sum
(6x + 5) dx
With upper limit 3 and lower limit 1

Answer 1

Answer:

∫₁³ (6x + 5) dx = 34

Step-by-step explanation:

I = ∫₁³ (6x + 5) dx

Let us divide the domain into n equal parts, each of length t. Then

t = (3-1)/n = 2/n

or n = 2/t

Considering each strip to be approximately rectangular, the above integral can be expressed as the following Upper Reimann Sum

I = lim ₜ→₀ [ {6(1+t) + 5}t + {6(1+2t) + 5}t +....+{6(1+nt) + 5}t

= lim ₜ→₀ [ 6nt + 6t²Σn + 5nt ]

= lim ₜ→₀ [ 11nt + 6t²n(n+1)/2]

= lim ₜ→₀ [ 11nt + 3t²n(n+1)]

= lim ₜ→₀ [ 11t*(2/t) + 3t²(2/t)(2/t+1)]

= lim ₜ→₀ [22 + 3*2(2+t)]

= 22 + 6(2+0)

= 22 + 12

= 34