Activity 6.1.3

In each of the following situations, our goal is to determine the area of the region described. For each region, (i) determine the intersection points of the curves, (ii) sketch the region whose area is being found, (iii) draw and label a representative slice, and (iv) state the area of the representative slice. Then, state a definite integral whose value is the exact area of the region, and evaluate the integral to find the numeric value of the region's area. Note well: At the step where you draw a representative slice, you need to make a choice about whether to slice vertically or horizontally.

1. The finite region bounded by $x=y^2$ and $x=6-2y^2$.

2. The finite region bounded by $x=1-y^2$ and $x = 2-2y^2$.

3. The area bounded by the $x$-axis, $y=x^2$, and $y=2-x$.

4. The finite regions between the curves $x=y^2-2y$ and $y=x$.