Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| (A) | The value of \(\int_{0}^{4}\displaystyle |x|\,dx\) is | (I) | 3 |
| (B) | The value of \(\int_{-2}^{2}\displaystyle |x|\,dx\) is | (II) | -1 |
| (C) | The value of \(\int_{0}^{3}\displaystyle [x]\,dx\) is | (III) | 8 |
| (D) | The value of \(\int_{-1}^{1}\displaystyle [x]\,dx\) is | (IV) | 4 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A)-(III), (B)-(IV), (C)-(I), (D)-(II)
Here \([x]\) is the greatest integer function, which is constant between consecutive integers, so its integral is found interval by interval.
(A) The value of \(\int_{0}^{4} |x|\,dx\) is
This matches (III) 8. On [0, 4], \(|x| = x\):
\( \displaystyle \int_{0}^{4} x\,dx = \left[\frac{x^2}{2}\right]_{0}^{4} = 8 \)
(B) The value of \(\int_{-2}^{2} |x|\,dx\) is
This matches (IV) 4. Since \(|x|\) is an even function:
\( \displaystyle \int_{-2}^{2} |x|\,dx = 2\int_{0}^{2} x\,dx = 2\left[\frac{x^2}{2}\right]_{0}^{2} = 4 \)
(C) The value of \(\int_{0}^{3} [x]\,dx\) is
This matches (I) 3. \([x] = 0, 1, 2\) on [0, 1), [1, 2), [2, 3):
\( \displaystyle \int_{0}^{3} [x]\,dx = \int_{0}^{1} 0\,dx + \int_{1}^{2} 1\,dx + \int_{2}^{3} 2\,dx \)
\(= 0 + 1 + 2 = 3\)
(D) The value of \(\int_{-1}^{1} [x]\,dx\) is
This matches (II) −1. \([x] = -1\) on [−1, 0) and \([x] = 0\) on [0, 1):
\( \displaystyle \int_{-1}^{1} [x]\,dx = \int_{-1}^{0} (-1)\,dx + \int_{0}^{1} 0\,dx = -1 \)
So the matching is (A)-(III), (B)-(IV), (C)-(I), (D)-(II).