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

Given curves \( x = y^2 - 2 \) and \( x = y \), the points of intersection are \( (-2, 0) \) and \( (2, 2) \). To find the area, we integrate the difference between the two functions over the range from \( y = -2 \) to \( y = 2 \): \[ A = \int_{-1}^{2} y \, dy - \int_{-1}^{2} (y^2 - 2) \, dy \]
\[= \left[\frac{y^2}{2} - \frac{y^3}{3} + 2y\right]_{-1}^{2} = \left(\frac{4}{2} - \frac{8}{3} + 4\right) - \left(\frac{1}{2} + \frac{1}{3} - 2\right)\]
\[= \frac{10}{3} + \frac{7}{6} = \frac{27}{6} = \frac{9}{2} \] Thus, the area is \( \frac{9}{7} \).