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 determine the area enclosed by the curves, an integral must be established. Initially, both curves will be expressed in terms of \( y \), and their intersection points will be calculated. The first curve is the parabola \( y = x^2 - 4x + 4 \), and the second is the sideways parabola \( y^2 = 16 - 8x \).

Subsequently, the intersection points are found by equating the two curves. The enclosed area is then calculated by integrating the difference between the two functions.

Upon completion of the integration, the resultant area enclosed by the curves is \( \frac{8}{3} \).