Question

The area of the region bounded by the curves \( x = y^2 - 2 \) and \( x = y \) is:

• A. \( \frac{9}{4} \)
• B. 9
• C. \( \frac{9}{2} \)
• D. \( \frac{9}{7} \)

Hint:

For finding the area between curves, set up an integral with the difference of the functions. Ensure the limits of integration are the points where the curves intersect. Simplify the integrand before computing the area.

Answer 1

The Correct Option is C

The intersection points of the curves \( x = y^2 - 2 \) and \( x = y \) are \( (-2, 0) \) and \( (2, 2) \).The area is calculated by integrating the difference between the functions over the interval \( y = -2 \) to \( y = 2 \):\[A = \int_{-1}^{2} y \, dy - \int_{-1}^{2} (y^2 - 2) \, dy \]
\[= \left[\frac{y^2}{2} - \frac{y^3}{3} + 2y\right]_{-1}^{2} = \left(\frac{4}{2} - \frac{8}{3} + 4\right) - \left(\frac{1}{2} + \frac{1}{3} - 2\right)\]
\[= \frac{10}{3} + \frac{7}{6} = \frac{27}{6} = \frac{9}{2}\]The resulting area is \( \frac{9}{7} \).