Final answer:
To sketch the region enclosed by the curves x = 9y² and x = 8 - 7y², plot both parabolas and find the intersection points by solving the equation 9y² = 8 - 7y². The enclosed region is between these curves from the leftmost to the rightmost points of intersection.
Step-by-step explanation:
To sketch the region enclosed by the curves x = 9y² and x = 8 - 7y², first plot each curve on the same coordinate system.
- For x = 9y², this represents a parabola that opens to the right with the vertex at the origin.
- For x = 8 - 7y², this represents a downward opening parabola that has been translated to the right by 8 units.
Find the points of intersection by equating the two expressions for x and solving for y:
9y² = 8 - 7y²
Combining like terms gives 16y² = 8, so y² = 1/2, which leads to y = ±1/√2.
Now, plot the points of intersection (4,1/√2) and (4,-1/√2) and sketch the two parabolas. The enclosed region is between these two parabolas from the leftmost point of intersection to the rightmost point of intersection, which are, in fact, the same point here because the curves are symmetric with respect to the x-axis.