Question

Find the exact area of the region Ω bounded by the curves x=−y²+5 and x=(y−1)² using horizontal rectangles.

Answer 1

To find the exact area of the region Ω bounded by the curves x = -y² + 5 and x = (y - 1)² using horizontal rectangles, we can partition the region into small rectangles and sum up their areas.

Step-by-step explanation:

To find the area of the region Ω bounded by the curves x = -y² + 5 and x = (y - 1)² using horizontal rectangles, we can partition the region into small horizontal rectangles and sum up their areas. Let's start by finding the intersection points of the curves:

1. Set the two equations equal to each other: -y² + 5 = (y - 1)²
2. Expand and simplify the equation: -y² + 5 = y² - 2y + 1
3. Rearrange the equation: 2y² - 2y - 6 = 0
4. Factor out a 2: 2(y² - y - 3) = 0
5. Factor the quadratic equation: 2(y + 1)(y - 3) = 0
6. Solve for y: y = -1 or y = 3

The intersection points are (-1, 0) and (3, 0). To find the width of each rectangle, we can take the difference between the x-coordinates of the intersection points at each y-value. The height of each rectangle is the change in y-value as we move from one rectangle to the next. Finally, we can calculate the area of each rectangle by multiplying its width and height. Adding up the areas of all the rectangles will give us the exact area of the region Ω.