Find the area of the surface generated when y=4x and x=1 is revolved about the y-axis.
No calculus need for this one. It's just a cylinder with a cone cut out.
r=1
h=4
v = πr^2h - π/3 r^2h = 2π/3 r^2h = 8π/3
Ok ok. If you want to use calculus, then with shells,
v = ∫[0,1] 2πrh dx
where r=x and h=y=4x
v = ∫[0,1] 2πx*4x dx = 8π/3
using discs (washers) you get
v = ∫[0,4] π(R^2-r^2) dy
where R=1 and r=x=y/4
∫[0,4] π(1-y^2/16) dy = 8π/3
We have to use the Surface area of revolution formula
integral(f(x) aqrt(1+f'(x)^2))
To find the area of the surface generated when y = 4x is revolved about the y-axis, we need to use the method of cylindrical shells.
The formula for calculating the area using cylindrical shells is:
A = 2π ∫(from a to b) (x * h) dx
Where:
- A is the area of the surface generated
- π is a constant, approximately 3.14159
- x is the variable of integration
- h is the height of each cylindrical shell
- a and b are the x-values that define the region of integration
Let's find the values of a and b by solving the equation y = 4x for x when y = 0:
0 = 4x
x = 0
So, a = 0.
To find the value of b, we need to solve the equation y = 4x for x when y = 4:
4 = 4x
x = 1
So, b = 1.
Now we can substitute the values into the formula:
A = 2π ∫(from 0 to 1) (x * h) dx
We still need to determine the height of each cylindrical shell, which is given by the equation h = 4x.
Substituting h = 4x into the formula, we get:
A = 2π ∫(from 0 to 1) (x * 4x) dx
Simplifying the equation:
A = 8π ∫(from 0 to 1) (x^2) dx
Now we can integrate:
A = 8π * [(x^3)/3] |(from 0 to 1)
Plugging in the upper and lower limits:
A = 8π * [(1^3)/3] - 8π * [(0^3)/3]
Simplifying further:
A = 8π * (1/3) - 8π * (0/3)
A = 8π/3
Therefore, the area of the surface generated when y = 4x is revolved about the y-axis is 8π/3 square units.