Express the limit as a definite integral on the given interval.

Question

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Answer 1

The definition of a definite integral in terms of sum and limits is given by:

∫^b_a f(x) dx = lim_n→∞ ∑ⁿ_i=1 f(x_i) Δx

Where Δx = (b-a)/n

x_i = a + Δx• i

The given limit is:

lim_n→∞ ∑ⁿ_i=1[ 5(x_i)³ - 5x_i ] Δx , [2,4]

That means:

f(x_i) = 5(x_i)³ - 5x_i

f(x) = 5x³ - 5x

Therefore our definite integral is:

∫⁴₂ (5x³ - 5x) dx