Find the volume formed by rotating the region enclosed by:
x=5y and y3=x with y0
To find the volume formed by rotating the region enclosed by the curves x = 5y and y³ = x, we can use the method of cylindrical shells.
First, let's sketch the curves to get an idea of the region we are working with.
The curve x = 5y can be written as y = x/5, and the curve y³ = x can be rewritten as x = y³.
So we have two curves:
1. y = x/5
2. x = y³
To find the bounds of integration, we need to determine where these curves intersect.
Setting y = x/5 equal to x = y³, we get:
x/5 = y³
Rearranging the equation, we have:
y³ = x/5
y³ = y²/5
5y³ = y²
Simplifying, we get:
5y³ - y² = 0
Factoring out y², we obtain:
y²(5y - 1) = 0
This gives us two solutions: y = 0 and y = 1/5.
Now we can set up the integral to find the volume.
The volume V is given by:
V = ∫[a,b] 2πrh dh
where [a,b] represents the bounds of integration in the y-direction, r is the distance between the axis of rotation (x = 0) and the curve x = 5y, and h is the height of the cylindrical shell.
In this case, the axis of rotation is the y-axis (x = 0), so the distance r is equal to y.
The height h is the difference between the x-coordinates of the curves x = 5y and y³ = x.
h = (5y) - (y³) = 5y - y³
Now we can set up the integral:
V = ∫[0,1/5] 2πy(5y - y³) dy
Evaluating this integral will give you the volume formed by rotating the region enclosed by the curves x = 5y and y³ = x.