The region bounded by the line y = x, the x-axis and the line x = 4 is divided into two regions of equal area by the line x = k. Then k is equal to
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(2\sqrt{2}\)
The whole region lies under y = x from x = 0 to x = 4:
\( \displaystyle \int_{0}^{4}x\,dx = \left[\frac{x^{2}}{2}\right]_{0}^{4} = 8 \)
The part to the left of x = k must have half of this area, that is 4:
\( \displaystyle \int_{0}^{k}x\,dx = \frac{k^{2}}{2} = 4 \)
\(k^{2} = 8,\ \ k = 2\sqrt{2}\) (since 0 < k < 4)
Why the other options are incorrect
Option (1) 2 — Incorrect. The part to the left of x = 2 has area \(\frac{4}{2} = 2\), only a quarter of 8; halving the base does not halve the area.
Option (2) \(\sqrt{2}\) — Incorrect. The part to the left of \(x = \sqrt{2}\) has area \(\frac{2}{2} = 1\).
Option (3) 3 — Incorrect. The part to the left of x = 3 has area \(\frac{9}{2}\), more than 4.