Question

Find the exact area of the region Ω bounded by the curves x=−y² +5 and x=(y−1)² using horizontal rectangles. As part of your work sketch the region on the axis below without using a graphing calculator or computer, draw an approximating rectangle, and start your work with dA 6.5. Do the same problem as #5 but use vertical rectangles. As part of your work explain why this would be longer than 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, divide the region into small horizontal strips and calculate the area of each strip. Using vertical rectangles would result in a longer process because the width of the rectangles would vary along the y-axis.

Step-by-step explanation:

To find the exact area of the region Ω bounded by the curves x = -y² + 5 and x = (y - 1)² using horizontal rectangles, you can divide the region into small horizontal strips and calculate the area of each strip. This can be done by multiplying the width of the strip (dA) by the value of x along that strip. Start with dA = 6.5 and calculate the area of each rectangle, then sum up all the areas to find the exact area of the region.

Using vertical rectangles to find the area of the region would result in a longer process compared to using horizontal rectangles because the width of the vertical rectangles would vary along the y-axis. This means that you would need to calculate the width of each rectangle for different values of y, making the process more complex and time-consuming.