Find the exact area of the surface obtained by rotating the curve about the x-axis.
y = sqrt(1+5x) from 1<x<7
The surface is generated by rotating the curve around the x-axis.
Take a vertical strip of width Δx.
the incremental area is 2πyΔs
where Δs = sqrt(1+(dy/dx)²).
let
y(x)=sqrt(1+5x),
y'(x)=5/(2sqrt(1+5x)
(y'(x))²=25/(4(1+5x))
So
Total surface area
7
=∫y*sqrt(1+y'²)dx
1
=∫sqrt(1+5x)sqrt(1+(5/(2sqrt(1+5x))^2)dx
=∫sqrt(20x+29)/2dx
=(20x+29)^(3/2)/60 from 1 to 7
=30.9
Check my arithmetic.
To find the exact area of the surface obtained by rotating the curve y = sqrt(1+5x) about the x-axis, we can use the method of integration.
Step 1: Express the curve in terms of x as a function of y.
To do this, we need to solve the equation for x.
y = sqrt(1 + 5x)
Squaring both sides, we get:
y^2 = 1 + 5x
Rearranging, we have:
5x = y^2 - 1
Dividing both sides by 5, we get:
x = (y^2 - 1)/5
So, the curve can be expressed as x = (y^2 - 1)/5.
Step 2: Determine the limits of integration.
Given that the curve is rotated about the x-axis from x = 1 to x = 7, we need to find the corresponding limits of y.
To find the limits of y, we substitute the x-values into the equation x = (y^2 - 1)/5.
For x = 1:
1 = (y^2 - 1)/5
5 = y^2 - 1
y^2 = 6
y = ±√6
For x = 7:
7 = (y^2 - 1)/5
35 = y^2 - 1
y^2 = 36
y = ±6
Since the curve is symmetric about the x-axis, we only need to consider the positive values of y. Therefore, the limits of integration for y are √6 to 6.
Step 3: Set up the integral for the surface area.
The formula for the surface area of a curve rotated about the x-axis is:
A = 2π ∫[a,b] f(x) √(1 + (f'(x))^2) dx
In this case, f(x) = sqrt(1 + 5x), and we need to express this in terms of y using the equation x = (y^2 - 1)/5 obtained earlier. We also need to find f'(x).
Taking the derivative of x = (y^2 - 1)/5, we have:
1/5 = (2y)(dy/dx)
dy/dx = 1/(10y)
Substituting these values into the surface area formula gives us:
A = 2π ∫[√6,6] [(y^2 - 1)/5] √(1 + (1/(10y))^2) dy
Step 4: Integrate to find the area.
Now, we can integrate the equation above with respect to y over the limits of integration, √6 to 6.
A = 2π ∫[√6,6] [(y^2 - 1)/5] √(1 + (1/(10y))^2) dy
Using the appropriate methods of integration, the final step is to evaluate this integral to find the exact area of the surface obtained by rotating the curve about the x-axis.