The area of the region bounded by the lines x + 2y = 12, x = 2, x = 6 and x-axis is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 16 sq units
From x + 2y = 12,
\( \displaystyle y = \frac{12-x}{2} = 6-\frac{x}{2} \)
For x between 2 and 6 this is positive, so the line lies above the x-axis there and the area is
\( \displaystyle A = \int_{2}^{6}\left(6-\frac{x}{2}\right)dx \)
\( \displaystyle = \left[6x-\frac{x^2}{4}\right]_{2}^{6} \)
\( = (36-9)-(12-1) \)
\( = 27-11 = 16 \) sq units
Shortcut: The region is a trapezium with parallel vertical sides y = 5 (at x = 2) and y = 3 (at x = 6), a distance 4 apart:
\( \displaystyle A = \frac{1}{2}(5+3)\times 4 = 16 \)