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NDA 2016
Exam
NDA
Year
2016
Subject
Mathematics
Chapter
Definite Integration

NDA 2016 · Mathematics · Definite Integration

What is ∫_ -2 ^2 x d x-∫_ -2 ^2[x] d x equal to, where [·] is the...

What is ∫−22xdx−∫−22[x]dx\int_{-2}^2 x d x-\int_{-2}^2[x] d x equal to, where [⋅][\cdot] is the greatest integer function?

Options

  1. A0
  2. B1
  3. C2
  4. D4

Correct answer

C. 2

Step-by-step solution

∫−22xdx−∫−22[x]dx=[x22]−22−∫−2−1[x]dx−∫−10[x]dx−∫01[x]dx−∫12[x]dx=12[4−4]−(−2)−(−1)−0−(1)=2+1−1=2\begin{aligned} & \int_{-2}^2 x d x-\int_{-2}^2[x] d x \\ & =\left[\frac{x^2}{2}\right]_{-2}^2-\int_{-2}^{-1}[x] d x-\int_{-1}^0[x] d x-\int_0^1[x] d x-\int_1^2[x] d x \\ & =\frac{1}{2}[4-4]-(-2)-(-1)-0-(1) \\ & =2+1-1=2\end{aligned}
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