find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. sketch the region and a typical disk or washer.
y^2=x, x=2y; about the y-axis
i am confused because i do not know how to set it up because one is Y and the other X i know the formula
The curves intersect at (0,0) and (4,2)
Imagine the washers around the y-axis.
We have
v = ∫[0,2] π(R^2-r^2) dy
where r = y^2 and R = 2y
v = π∫[0,2] (2x)^2 - (y^2)^2 dy
= 64/15 π
Just to check, using shells, we have
v = ∫[0,4] 2πrh dx
where r=x and h = √x - x/2
v = 2π∫[0,4] x(√x - x/2) dx
= 64/15 π
To find the volume of the solid obtained by rotating the region bounded by the curves y^2 = x and x = 2y about the y-axis, you can use the method of cylindrical shells. Here's how you can set it up:
Step 1: Sketch the region and a typical disk or washer:
Draw the curves y^2 = x and x = 2y on a coordinate plane to visualize the region. The curve y^2 = x is a parabola opening to the right, while x = 2y is a straight line. The region bounded by these curves will be enclosed between the two curves. Next, draw a typical disk or washer within this region. A typical disk in this case will be a cylindrical shell along the y-axis, with a radius equal to the value of y and a height equal to the difference between the x-values of the curves at that y-value.
Step 2: Determine the limits of integration:
To set up the integral for finding the volume, you need to determine the limits of integration. From the given information, we know that x = 2y, so we can solve this equation to find the y-values where the curves intersect. Substitute x = 2y into the equation y^2 = x:
y^2 = 2y
y^2 - 2y = 0
y(y - 2) = 0
So, the y-values where the curves intersect are y = 0 and y = 2. Hence, the limits of integration for y will be from y = 0 to y = 2.
Step 3: Set up the integral for volume using cylindrical shells:
The volume element of a cylindrical shell can be expressed as dV = 2πrhdy, where r is the radius of the cylindrical shell (in this case, r = y) and h is the height (in this case, h = x = 2y). Integrate this volume element over the limits of y to find the total volume:
∫(from 0 to 2) 2π(2y)(2y) dy
Simplify the equation:
∫(from 0 to 2) 8πy^2 dy
Integrate with respect to y:
[ (8π/3)y^3 ] (from 0 to 2)
Evaluate the integral at the limits of integration:
(8π/3)(2^3) - (8π/3)(0^3)
(8π/3)(8) - (8π/3)(0)
64π/3 cubic units
Therefore, the volume of the solid obtained by rotating the region bounded by the curves y^2 = x and x = 2y about the y-axis is 64π/3 cubic units.