Final answer:
To calculate the volume of the solid created by revolving the region between y = x and y = x^2, the shell and washer methods are used. For the shell method, integrate 2π(radius)(height)(thickness), and for the washer method, integrate π(outer radius)^2 - π(inner radius)^2, for both x and y-axes.
Step-by-step explanation:
To compute the volume of the solid generated by revolving the region bounded by the curves y = x and y = x^2 about each coordinate axis, we can use both the shell method and the washer method.
Shell Method
Revolving about the y-axis: The volume is found by integrating 2π(radius)(height)(thickness) where the radius is 'y' (since y = x), the height is 'x - x^2' (since it's bounded by y = x and y = x^2), and the thickness is dy. Integrate from y = 0 to y = 1.
Revolving about the x-axis: The volume is found by integrating 2π(radius)(height)(thickness) where the radius is 'x', the height is 'x - x^2', and the thickness is dx. Integrate from x = 0 to x = 1.
Washer Method
Revolving about the y-axis: The volume is found by integrating π(outer radius)^2 - π(inner radius)^2 (the area of a washer) dx, where the outer radius is 'x' and the inner radius is x^2. Integrate from x = 0 to x = 1.
Revolving about the x-axis: The volume is found by integrating π(outer radius)^2 - π(inner radius)^2 dy, where the outer radius is 'x' and the inner radius is y. Integrate from y = 0 to y = 1.