Sketch the region bounded by the graphs of the given equations and find the area of their region. x = y² + 2, x = y - 2, y = -3, y = 3

Question

Quizplus · Accepted Answer

The given equations are:

x = y^2 - y + 2 
x = y - 2 
y = -3 
y = 3

We can start by plotting the vertical lines x = -2 and x = 2, which are the intercepts of the line x = y - 2 with the x-axis:

- When x = -2, y = -4 and y = 0 (by solving x = y - 2 and x = y^2 - y + 2) , so the point (-2,-4) lies on the parabola and the point (-2,0) on the line. 
- When x = 2, y = 1 and y = 3 (by solving x = y - 2 and x = y^2 - y + 2), so the point (2,1) lies on both curves and the point (2,3) lies on the other horizontal line.

Therefore, the region of interest is a rectangle with height 6 and base 4, so its area is:

A = 6 * 4 = 24

Thus, the correct answer is B.

Sketch the Region Bounded by the Graphs of the Given