Question

The area of the region enclosed by the curves \( y = x^2 - 4x + 4 \) and \( y^2 = 16 - 8x \) is:

• A. \( \frac{4}{3} \)
• B. \( 8 \)
• C. \( \frac{8}{3} \)
• D. \( 5 \)

Hint:

When finding the area between curves: - Set up the appropriate integral by first determining the points of intersection. - Integrate the difference between the two curves over the appropriate interval. - Always check the limits of integration and the nature of the curves involved.

Answer 1

The Correct Option is C

To find the area enclosed by the curves, we need to set up an integral. First, let's express both curves in terms of \( y \) and solve for the intersection points. The first curve is \( y = x^2 - 4x + 4 \) (a parabola) and the second curve is \( y^2 = 16 - 8x \), which is a sideways parabola. Next, we solve for the intersection points by equating the two curves, and then integrate the difference of the two functions to find the enclosed area. After solving the integration, the area enclosed by the curves is \( \frac{8}{3} \).