The area (in sq. units) of the region bounded by the line y = 2x, the x-axis and the lines x = −1 and x = 3 is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 10
The line y = 2x meets the x-axis at x = 0. It is below the x-axis for x < 0 and above it for x > 0, so the area is found in two parts and the part below the axis is taken with its absolute value.
\( \displaystyle \int_{-1}^{0} 2x\,dx = \left[x^{2}\right]_{-1}^{0} = 0-1 = -1 \)
\( \displaystyle \int_{0}^{3} 2x\,dx = \left[x^{2}\right]_{0}^{3} = 9 \)
So the area is
\(A = |-1| + 9 = 10\)
Why the other options are incorrect
Option (1) 8 — Incorrect. This is the single integral from −1 to 3, which is −1 + 9; the part below the axis is subtracted instead of added.
Option (2) 9 — Incorrect. This is only the part from x = 0 to x = 3.
Option (3) 1 — Incorrect. This is only the part from x = −1 to x = 0.