Find the volume of the solid obtained by revolving the graph of y=x(sqrt(1-x^2)) over [0,1] about the y-axis.

How do you write the integral?

Concentric rings of circumference 2 pi x and height y

integral 2 pi x y dy from 0 to 1

integral 2 pi x y dx from 0 to 1

To write the integral that represents the volume of the solid obtained by revolving the graph of y = x(sqrt(1 - x^2)) over [0, 1] about the y-axis, you can use the method of cylindrical shells.

The formula to find the volume using cylindrical shells is:
V = 2π ∫ [a, b] (x * h(x)) dx

In this case, the limits of integration are [a, b] = [0, 1], and the height of each cylindrical shell is h(x) = 2πx.

Plugging these values into the formula, we get:
V = 2π ∫ [0, 1] (x * 2πx) dx

Simplifying the equation further:
V = 4π^2 ∫ [0, 1] (x^2) dx

Now, you can integrate the function x^2 with respect to x over the interval [0, 1] and evaluate the integral to find the volume of the solid.