Find the area of the surface generated when y=4x and x=1 is revolved about the y-axis.
We have to use the surface area formula.
surface area formula? There are probably thousands of them. Cal 2, I assume you are doing integration.
However, the surface area formula of a cone strikes me quickly. Here you have a base cone of radius 4, and altitude of 1.
SA=PI*r (r+ sqrt(r^2+h^2))
Oh yeah. Sorry I did the volume instead.
Consider the surface as a pile of circular rings, each with circumference 2πr. So, the surface area is
S = ∫[0,1] 2πx ds
but ds is a small piece of the line, of length ds^2 = dx^2+dy^2 = dx^2 + (4dx)^2 = 5dx^2
S = ∫[0,1] 2πx√5 dx = √5 π
This is the lateral area as given by bobpursley above. He also included the base of the cone, but we do not want that for the surface of revolution.
To find the area of the surface generated when the line y=4x is revolved about the y-axis, we can use the method of cylindrical shells.
The formula for the surface area of a cylindrical shell is:
dA = 2πrh
Where dA represents the surface area of the cylindrical shell, r represents the radius, and h represents the height.
In this case, since we are revolving the line y=4x about the y-axis, the radius (r) of each cylindrical shell will simply be the x-coordinate of the point on the line at a given height.
To find the height (h) of each cylindrical shell, we need to consider the difference in the y-coordinates of two points on the line at a given x-coordinate (which represents the change in y as we move along the x-axis).
So let's break down the problem into smaller steps:
Step 1: Find the limits of integration. In this case, since the line passes through the origin (0, 0), we can integrate from x=0 to x=1.
Step 2: Calculate the radius (r) and height (h) of each cylindrical shell at a given x-coordinate.
Since the line equation is y=4x, the radius (r) at any x-coordinate will simply be equal to x.
The height (h) at a given x-coordinate can be found by subtracting the y-coordinate at that x-coordinate (which is 4x) from the y-coordinate at the x-coordinate of the previous cylindrical shell.
Step 3: Calculate the surface area of each cylindrical shell using the formula dA = 2πrh.
Step 4: Integrate the surface area over the limits of integration using the definite integral ∫dA from x=0 to x=1 to find the total surface area of the generated surface.
I can perform the calculations for you if you'd like.