Many dimensions

Calculus Level 2

What is

∫ 0 1 ∫ 0 1 ⋯ ∫ 0 1 ( x 1 + x 2 + ⋯ + x n ) d x 1 d x 1 ⋯ d x n \displaystyle \int_0^1 \int_0^1 \cdots \int_0^1 (x_1+x_2+\cdots+x_n) \, dx_1 dx_1 \cdots dx_n

for n = 2018 n=2018 ?


The answer is 1009.

Henry U by Henry U , 17, Germany

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1 solution

Parth Sankhe
Nov 2, 2018

We can integrate each variable separately, each of whose definite integral value would be 1 2 \frac {1}{2} . Thus the answer would be 2018 ⋅ ½ = 1009 2018\cdot½=1009

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