The area of the region bounded by the curve \( x=2y+3 \), y-axis and the lines \( y=-1 \) and \( y=1 \) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 6 Sq. units
The curve gives x in terms of y, so the area is taken with respect to y. Between y = −1 and 1, x = 2y + 3 stays positive:
\( \displaystyle \int_{-1}^{1}(2y+3)\,dy=\left[y^{2}+3y\right]_{-1}^{1} \)
\( \displaystyle =(1+3)-(1-3) \)
\( \displaystyle =6 \)
Shortcut: The region is a trapezium with parallel sides 1 and 5 (the values of x at y = −1 and y = 1) and height 2:
\( \displaystyle \frac{1}{2}(1+5)\cdot 2=6 \)