Area of region bounded by the curves \(x = y^3, x = 0\) between \(y = -1\) and \(y = 2\) is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{17}{4}\) sq. units
The curve \(x = y^3\) lies to the left of the y-axis (x = 0) for y < 0 and to the right of it for y > 0. Using horizontal strips, the area is \(\int |x|\,dy\), so the integral is split at y = 0:
\(\displaystyle \text{Area} = \int_{-1}^{0} (0 - y^3)\,dy + \int_{0}^{2} y^3\,dy\)
\(\displaystyle = \left[-\frac{y^4}{4}\right]_{-1}^{0} + \left[\frac{y^4}{4}\right]_{0}^{2}\)
\(\displaystyle = \frac{1}{4} + 4 = \frac{17}{4}\) sq. units
Tip: Integrating \(y^3\) straight from −1 to 2 gives \(4 - \frac{1}{4} = \frac{15}{4}\), Option (1): the part to the left of the y-axis is then subtracted instead of added.