I'm so confused

Question

Express the integral as a limit of Riemann sums. Do not evaluate the limit. (Use the right endpoints of each subinterval as your sample points.)

 ∫6 to 3 x/2+x^5dx

Answer 1

Δx = width of each interval = (6-3)/n = 3/n and f(x) = (1/2)x + x⁵

x₁ = right endpoint of first interval = 3 + 3/n

x₂ = right endpoint of second interval = 3 + 2(3/n)

x₃ = right endpoint of third interval = 3 + 3(3/n)... x_n = 3 + n(3/n) = 6

∫_{(from 3 to 6)} f(x)dx = lim_n→∞ [∑_{(i= 1 to n)} f(x_i)Δx] = (3/n)lim_n→∞ [∑_{(i= 1 to n)} ( (1/2)x_i + (x_i)⁵ )] =

(3/n)lim_n→∞ [∑_{(i= 1 to n)} ( (1/2)(3 + 3i/n) + (3 + (3i/n))⁵)]

Answer 2

Recall that the integral is defined as n-many little Riemann slices of width (b-a)/n and height f(x)

Therefore, your summation notation should look like:

n

lim (6-3)/n * ∑ (x_i/2 + x_i^5)dx

n→∞ i=1

Se this image for clarity https://slideplayer.com/slide/7639486/

If the notation used above is new to you, send me a message and I'll happily get you up to speed with your calculus prerequisite knowledge.

Rael