Question

Is the function \(\mathrm{f}(\mathrm{x})=\mathrm{e}^{-1 / \mathrm{x}}\) integrable on the closed intervals (i) \([-3,-2]\), (ii) \([-1,0]\) and (iii) \([-1,1] ?\)

Short answer

Yes, the function is integrable on all three closed intervals, as it is continuous on each interval away from \(x=0\).

Step-by-step solution

Check the continuity of the function on each interval

The function \(f(x) = e^{-\frac{1}{x}}\) has a discontinuity at \(x=0\). Therefore, we need to check the continuity of the function on the intervals away from 0. Since all of the given intervals do not include \(x=0\), the function will be continuous on these intervals.

Determine the integrability of the function on each interval

Since the function is continuous on each interval (away from \(x=0\)), it is automatically integrable on these intervals. Thus, the function \(f(x) = e^{-\frac{1}{x}}\) is integrable on: (i) \([-3, -2]\) (ii) \([-1, 0]\) (iii) \([-1, 1]\)