Final answer:
Using the method of cylindrical shells, the volume of the region bounded by y = x² and x = √y about the y-axis is 0.2π.
Step-by-step explanation:
To find the volume of the region bounded by y = x² and x = √y about the y-axis, we can use the method of cylindrical shells. We will integrate the volume of each shell from y = 0 to y = 1. The radius of each shell is given by x = √y, and the height of each shell is given by the difference between the curves y = x² and x = √y, which is (x² - √y).
The volume of each shell can be calculated as V = 2πrH, where V is the volume, r is the radius, and H is the height. Substituting the values, we have V = 2π√y(y - √y). Now, we integrate this expression with respect to y from 0 to 1 to find the total volume of the region.
∫(2π√y(y - √y))dy = 2π(∫(y^(3/2) - y)dy)
Using the power rule, we can integrate each term separately:
2π(∫y^(3/2)dy - ∫ydy) = 2π(2/5y^(5/2) - y^2/2)
Now, substitute the limits of integration and evaluate the expression:
2π((2/5(1)^(5/2) - (1)^2/2) - (2/5(0)^(5/2) - (0)^2/2)) = 2π((2/5 - 1/2) - (0)) = 2π(4/10 - 5/10) = 2π(-1/10) = -0.2π
Since the volume cannot be negative, the answer is 0.2π.