The area of the region bounded by \( x=y^{3} \), the y-axis and the lines \( y=-1 \) and \( y=2 \) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \displaystyle\frac{17}{4} \) sq.unit
Since the boundary is the y-axis, we integrate with respect to y. The curve \( x=y^{3} \) lies left of the y-axis for \( y\lt 0 \), so that part is taken with a minus sign:
\( \displaystyle \text{Area}=\int_{-1}^{0}(-y^{3})\,dy+\int_{0}^{2}y^{3}\,dy \)
\( \displaystyle =\frac{1}{4}+4 \)
\( \displaystyle =\frac{17}{4} \)
Tip: Integrating straight from −1 to 2 gives \( \int_{-1}^{2}y^{3}\,dy=\frac{15}{4} \) (Option 1), which is wrong because the part left of the y-axis then counts as negative area.