Area Between Two Curves

We are given two curves: The area enclosed between these curves can be found by integrating the difference of the two curves over the interval to . First, observe that will be equivalent to in the interval to , since cosine is non-negative in the given range. The area is given by: Now, integrate: Thus, the correct answer is option (4), .

The given curves are and . These represent two parabolas opening upwards. To find the area between these curves, we first solve for in terms of for both equations: The region of interest is the area between these two curves from to , because the curves intersect at (i.e., the point where both parabolas touch the X-axis). The area is given by: Simplifying the expression: Expanding the terms inside the integral: Now, integrate: Substitute the limits: Thus, the area between the curves is square units.