Use the disk method to find the volume of the solid formed by rotating the region bounded by y=2x and y=x^2 about the y-axis
The curves intersect at (0,0) and (2,2)
v = ∫[0,2] π(R^2-r^2) dy
where R = √y and r = y/2
v = ∫[0,2] π(y-y^2/4) dy
= π(y^2/2 - y^3/12) [0,2]
= 4π/3
To find the volume of the solid formed by rotating the region bounded by the functions y=2x and y=x^2 about the y-axis, we can use the disk method. The disk method involves slicing the solid into infinitesimally small disk-shaped cross-sectional slices, summing up the volumes of these tiny disks, and integrating them to find the total volume.
To use the disk method, we need to express the equations in terms of the variable of integration, which in this case is y. Let's rearrange the equations to solve for x:
For y = 2x:
x = y/2
For y = x^2:
x = √y
Now, we can determine the boundaries for our integral. The region is bounded by these two curves, so we need to find the intersection points:
2x = x^2
x^2 - 2x = 0
x(x - 2) = 0
This equation gives us two intersection points:
x = 0 and x = 2
Since we are rotating about the y-axis, the axis of rotation is the vertical line x = 0. Therefore, we will integrate with respect to y from the lower bound y = 0 to the upper bound y = 2, which corresponds to the intersection points.
Now, let's focus on finding the radius of each disk. Since we are rotating the region about the y-axis, the radius is the distance from the axis of rotation (y-axis) to the curve. In this case, the radius is given by:
Radius = x = √y (using the equation x = √y)
Next, we need to find the area of each disk. The formula for the area of a disk is:
Area = π * (radius)^2
Substituting the expression for the radius, we get:
Area = π * (√y)^2 = π * y
Now, we can set up the integral to find the volume:
Volume = ∫[0 to 2] π * y dy
Integrating with respect to y, we get:
Volume = π * ∫[0 to 2] y dy
Evaluating the integral, we get:
Volume = π * [y^2/2] from 0 to 2
= π * [(2^2)/2 - (0^2)/2]
= π * [4/2 - 0/2]
= π * 2
Therefore, the volume of the solid formed by rotating the region bounded by y=2x and y=x^2 about the y-axis is 2π cubic units.