Let A be the region bounded by the curves y = x^2-6x + 8 and y = 0.
Find the volume obtained when A is revolved around the Y-AXIS
To find the volume obtained when region A is revolved around the y-axis, we can use the method of cylindrical shells. This involves integrating the height of each cylindrical shell to obtain the total volume.
First, let's find the points where the curves intersect. We need to solve the equation x^2 - 6x + 8 = 0 for x.
x^2 - 6x + 8 = (x - 2)(x - 4) = 0
So, x = 2 and x = 4 are the x-coordinates of the points of intersection.
Now, we integrate to find the volume. The formula for the volume of a cylindrical shell is:
Volume = ∫(2πx * f(x)) dx
Where f(x) is the height of the cylindrical shell at x and 2πx represents the circumference of the shell.
To find the height, subtract the smaller function (y = 0) from the larger function (y = x^2 - 6x + 8):
f(x) = (x^2 - 6x + 8) - 0 = x^2 - 6x + 8
The integral becomes:
Volume = ∫(2πx * (x^2 - 6x + 8)) dx
Evaluate the integral from x = 2 to x = 4:
Volume = ∫[2,4] (2πx * (x^2 - 6x + 8)) dx
Now, you can use calculus or an online integral calculator to evaluate the integral to get the final volume value.