Find the volume of the solid given by rotating the region bounded by the curves y=x^2, x=1, x=2, and y=0 around the y-axis

a) Use the shell method

b) Use the washer method. Be careful with the radius of the washer at different y.

shells of thickness dx:

v = ∫[1,2] 2πrh dx
where r=x and h=y=x^2
v = ∫[1,2] 2πx*x^2 dx = 15π/2

washers of thickness dy -- the curved portion plus a cylinder 1 unit high with radii 1 and 2:

v = 3π + ∫[1,4] π(R^2-r^2) dy = 15π/2
where R=2 and r=x=√y
v = ∫[1,4] π(4-y) dy =

To find the volume of the solid generated by rotating the given region around the y-axis, we can use two different methods - the shell method and the washer method.

a) Shell Method:
The shell method involves summing up the volumes of infinitely thin cylindrical shells. Each shell has a radius obtained from the distance between the y-axis and a given x-value on the curve, and a height equal to the difference between the y-values of the curve at the top and bottom boundaries of the region.

To apply the shell method, consider a horizontal shell at a height y (between 0 and 1) and an infinitesimally small width dx. The radius of this shell is x (between 1 and 2). The height of the shell can be calculated as the difference between the y-values of the function at x and the bottom boundary of the region (y=0), which is x^2.

The volume of this shell is given by:
dV = 2πx(x^2)dx

To find the total volume, integrate this expression with respect to x from x=1 to x=2:
V = ∫(1 to 2) 2πx(x^2)dx

Evaluate this integral to get the volume.

b) Washer Method:
The washer method involves integrating the difference between the volumes of two concentric cylindrical washers. Each washer represents a small volume element obtained between two cross-sections perpendicular to the axis of rotation.

To apply the washer method, consider a horizontal washer at a height y (between 0 and 1) and an infinitesimally small width dy. The outer radius (R) of this washer is given by x=2, and the inner radius (r) is given by x=1. Since the region is rotated around the y-axis, the corresponding x-values depend on y and can be solved as x = √y.

The width of the washer can be calculated as the difference between the outer and inner radii:
width = R - r = 2 - 1 = 1

The height of the washer can be calculated as the difference between the y-values of the curve at the top and bottom boundaries of the region, which is given by:
height = x^2 - 0 = (√y)^2 = y

The volume of the washer is given by:
dV = π(R^2 - r^2)dy

Substituting the values in:
dV = π((2)^2 - (1)^2)dy
dV = π(3)dy

To find the total volume, integrate this expression with respect to y from y=0 to y=1:
V = ∫(0 to 1) π(3)dy

Evaluate this integral to get the volume.

By following these steps and performing the necessary calculations, you can find the volume of the solid using both the shell method and the washer method.