Find the area of the bounded region by the graph of x=2y-3 and x=y^2-2y in two ways
(i) using x axis as refrence y axis
(ii) using y axis as refrence x axis
(i) To find the area of the bounded region using the x-axis as the reference axis, we need to determine the x-coordinates of the points at which the two curves intersect.
1. Set the two equations equal to each other:
2y - 3 = y^2 - 2y
2. Rearrange the equation to obtain a quadratic equation:
y^2 - 4y + 3 = 0
3. Factorize the quadratic equation:
(y - 1)(y - 3) = 0
Therefore, y = 1 or y = 3.
4. Substitute the y-values back into either equation to obtain the corresponding x-values:
For y = 1:
x = 2(1) - 3 = -1
For y = 3:
x = 2(3) - 3 = 3
5. Now, integrate the absolute difference between the curves with respect to x from -1 to 3 to find the area:
Area = ∫|x - (y^2 - 2y)| dx, limits: -1 to 3
6. Substitute x = 2y - 3 into the equation to simplify:
Area = ∫|(2y - 3) - (y^2 - 2y)| dx, limits: -1 to 3
7. Evaluate the integral to obtain the area.
(ii) To find the area of the bounded region using the y-axis as the reference axis, we need to determine the y-coordinates of the points at which the two curves intersect.
1. Set the two equations equal to each other:
2y - 3 = y^2 - 2y
2. Rearrange the equation to obtain a quadratic equation:
y^2 - 4y + 3 = 0
3. Factorize the quadratic equation:
(y - 1)(y - 3) = 0
Therefore, y = 1 or y = 3.
4. Substitute the y-values back into either equation to obtain the corresponding x-values:
For y = 1:
x = 2(1) - 3 = -1
For y = 3:
x = 2(3) - 3 = 3
5. Now, integrate the difference between the curves with respect to y from 1 to 3 to find the area:
Area = ∫((y^2 - 2y) - (2y - 3)) dy, limits: 1 to 3
6. Simplify the equation:
Area = ∫(y^2 - 4y + 3) dy, limits: 1 to 3
7. Evaluate the integral to obtain the area.
By following these steps, you can find the area of the bounded region using both the x-axis and y-axis as reference axes.