The curve y=sinh(x),0<=x<=1, is revolved about the x-axis. Find the area of the resulting surface.
by definition
sinh(x) = (e^x - e^-x)/2
so the area
= ∫sinh(x) dx from 0 to 1
= ∫(e^x - e^-x)/2 dx from 0 to 1
= [ e^x + e^-x)/2] from 0 to 1
= (e^1 + e^-1)/2 - (e^0 + e^0)/2
= (e + 1/e)/2 - (1+1)/2
= e/2 + 1/(2e) - 1
= (e^2 + 1 - 2e)/(2e)
= (e - 1)^2 / (2e)
you better check my arithmetic
scrap my reply,
I found the area, silly me
To find the area of the surface formed by revolving the curve y = sinh(x) about the x-axis, you can use the formula for the surface area of a curve revolved around the x-axis.
The formula for the surface area, also known as the Euler's spiral formula, is given by:
A = 2π ∫[a,b] y * √(1 + (dy/dx)^2) dx
In this case, the function y = sinh(x) is the curve to be revolved, and the limits of integration are 0 and 1.
To calculate the surface area, you need to first find the derivative of y with respect to x, which is dy/dx. Then, substitute the values of y, dy/dx, and x into the formula and integrate with respect to x over the given range.
Let's go through the steps:
Step 1: Find dy/dx
To find dy/dx, take the derivative of y = sinh(x) with respect to x:
dy/dx = cosh(x)
Step 2: Substitute the values into the formula
The surface area formula becomes:
A = 2π ∫[0,1] sinh(x) * √(1 + cosh(x)^2) dx
Step 3: Evaluate the integral
To evaluate the integral, you can use various integration techniques such as substitution or trigonometric identities. After integrating, you will have a numerical value for the surface area.
Note that this integral might not have a closed-form solution, so you might need to use numerical methods or approximation techniques to find the answer.
Revolved about the x-axis~ must use: integral 2*Pi*y ds (Eq 1)
For ds use:
ds = sqrt(1+[F'(x)]^2) dx (Eq 2)
sinh(x) dx = cosh(x) = F'(x)
sub cosh(x) into Eq 2 and you get:
ds = sqrt(1+[cosh(x)]^2) dx
now sub ds into Eq 1
integral 2*Pi*y*sqrt(1+[cosh(x)]^2) dx
We are taking the derivative with respect to x so change y to it's equivalent.
integral 2*Pi*sinh(x)*sqrt(1+[cosh(x)]^2) dx, from 0 to 1
you now have your final equation to integrate over the interval, 0 to 1
and you get 5.52989 (final answer)