Find the volume of the solid obtained by rotating the region bounded by y=x and y=√x about the line x=2.
Volume =
intersection:
√x = x
x = x^2
x = 0 or x = 1
V = π∫ (2-√x)^2 - (2-x)^2 dy from y = 0 to 1
expand, integrate, etc
Now, you may be asking yourself, "Self, how can I integrate
∫ (2-√x)^2 dy ?"
You just need to express x as a function of y. That makes the volume
v = π∫[0,1] (2-y^2)^2 - (2-y)^2 dy = 8π/15
Thanks for the correction, oobleck
To find the volume of the solid obtained by rotating the region bounded by y=x and y=√x about the line x=2, we can use the method of cylindrical shells.
First, let's sketch the region bounded by y=x and y=√x:
The graph of y=x is a straight line passing through the origin with a slope of 1. The graph of y=√x is a curve starting at the origin and increasing as x increases.
To find the points of intersection between y=x and y=√x, we can set them equal to each other and solve for x:
x = √x
Squaring both sides, we get:
x^2 = x
Subtracting x from both sides, we have:
x^2 - x = 0
Factoring out x, we get:
x(x - 1) = 0
So, x = 0 or x = 1.
Therefore, the region bounded by y=x and y=√x is between x=0 and x=1.
Next, let's consider a vertical strip (or a cylindrical shell) within this region, with thickness Δx. The height of this strip is given by the difference between y=x and y=√x, which is (x - √x).
The radius of the cylindrical shell is the distance from the line x=2 to the axis of rotation, which is 2 - x.
The volume of this cylindrical shell can be approximated by multiplying the height, the circumference (2π), and the thickness Δx:
Volume of one cylindrical shell = 2π(x - √x)(2 - x)Δx
To find the total volume, we need to sum up the volumes of all the cylindrical shells from x=0 to x=1. This can be done by integrating the expression for the volume:
Total Volume = ∫[0,1] 2π(x - √x)(2 - x) dx
Now, we can evaluate this integral using calculus techniques, such as integration by substitution or integration by parts, to find the exact value of the volume.