Final answer:
To sketch the region enclosed by the given curves, find the points of intersection between the two equations and determine the region's boundaries. To find the area of the region, integrate with respect to either x or y using the formula for the area between two curves.
Step-by-step explanation:
To sketch the region enclosed by the given curves, you need to find the points of intersection between the two equations. Setting the equations equal to each other, you get 18 - 2y² = 2y² - 18. Simplifying this equation, you get 4y² = 36, or y² = 9. Taking the square root, you get y = ±3. Substituting these values back into one of the equations, you get x = 9 and x = -9.
The region enclosed by the curves is a parabolic shape between the points (9, 3) and (-9, -3). To find the area of this region, you can integrate with respect to x or y. Since the equation y = 2y² - 18 is already solved for y, it would be more convenient to integrate with respect to y.
Using the formula for the area between two curves, A = ∫(upper curve - lower curve), from y = -3 to y = 3, you can integrate the equation x = 18 - 2y² - (2y² - 18) with respect to y. This will give you the area of the region enclosed by the curves.