Let f: R - R be defined as: f(x) = |x - [x]|, if [x] is odd; f(x) = |x - [x+2]|, if [x] is even. If K = integral from -1 to 2 of f(x) dx, find the value of [K] (greatest integer less than or equal to K).

Question

Let f: R -> R be defined as: f(x) = |x - [x]|, if [x] is odd; f(x) = |x - [x+2]|, if [x] is even. If K = integral from -1 to 2 of f(x) dx, find the value of [K] (greatest integer less than or equal to K).

Accepted Answer

1. The integral is from -1 to 2. Wait, question says 1 to 2 originally but I'll use -1 to 2 as that makes more sense for this function. Let me re-read: 'integral from 1 to 2' - with only [1,2) where [x]=1 (odd): f(x)=|x-1|=x-1. Integral from 1 to 2 of (x-1)dx = [x²/2 - x] from 1 to 2 = (2-2)-(1/2-1) = 0+1/2 = 1/2. K=1/2, [K]=0. But if integral is from -1 to 2: On [-1,0): integral of (x+1)dx = [x²/2+x] = (0)-(1/2-1)=1/2. On [0,1): integral of (2-x)dx = [2x-x²/2] from 0 to 1 = 2-1/2 = 3/2. On [1,2): integral of (x-1)dx = 1/2. Total K=1/2+3/2+1/2=5/2. [K]=2.

Let f: R -> R be defined as: f(x) = |x - [x]|, if [x] is odd; f(x) = |x - [x+2]|, if [x] is even. If K = integral from -1 to 2 of f(x) dx, find the value of [K] (greatest integer less than or equal to K).

Solution

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