Find the volume V of the solid obtained by rotating the region bounded
by curves y=x and y= √𝒙 about the x-axis.
the curves intersect at (0,0) and (1,1)
using shells of thickness dy,
v = ∫[0,1] 2πrh dy
where r = y and h = y - y^2
v = ∫[0,1] 2πy(y-y^2) dy = π/6
using discs (washers) of thickness dx,
v = ∫[0,1] π(R^2-r^2) dx
where R = √x and r=x
v = ∫[0,1] π(x-x^2) dx = π/6
To find the volume V of the solid obtained by rotating the region bounded by the curves y = x and y = √x about the x-axis, you can use the method of cylindrical shells.
Step 1: Determine the limits of integration.
We need to find the x-values where the two curves intersect. Set y = x equal to y = √x and solve for x.
x = √x
Square both sides: x^2 = x
Rearrange: x^2 - x = 0
Factor out x: x(x - 1) = 0
Then x = 0 or x = 1.
So the limits of integration are x = 0 and x = 1.
Step 2: Find the height of each shell.
Since we are rotating the region about the x-axis, the height of each shell will be the difference between the y-values of the two curves evaluated at a particular x-value.
For a given x, the difference in height is: h = √x - x.
Step 3: Find the circumference of each shell.
The circumference of each cylindrical shell is given by the formula: C = 2πr, where r is the distance from the axis of rotation to the shell.
In this case, the distance from the axis of rotation (x-axis) to the shell is simply the x-value itself.
So the circumference of each shell is: C = 2πx.
Step 4: Set up the integral.
The volume of each cylindrical shell is given by the formula: dV = C * h * dx.
Integrating this expression with the appropriate limits of integration will give us the total volume.
V = ∫(from 0 to 1) 2πx(√x - x) dx
Now you can evaluate this integral to find the volume V.