The volume of the solid obtained by rotating the region enclosed by
y=1/x^4 y=0 x=2 x=4
about the line x=−5 can be computed the method of cylindrical shells via an integral
The volume is V
To find the volume of the solid using the cylindrical shell method, you can use the following integral formula:
V = ∫(2π * r * h)dx
where r represents the distance from the line of rotation (x = -5) to the function y(x), and h represents the height of an infinitesimally thin cylindrical shell.
First, let's find the expression for r and h.
Since we are rotating the region about the line x = -5, the distance to the function y(x) is x + 5.
The height of an infinitesimally thin cylindrical shell can be found by taking the difference between the upper and lower y-values, which in this case are y = 1/x^4 and y = 0.
So, the expression for r is: r = x + 5
And the expression for h is: h = 1/x^4 - 0 = 1/x^4 = x^(-4)
Plugging these values into the formula, we have:
V = ∫(2π * (x + 5) * (x^(-4)))dx
Now, we need to set up the integration limits. The region is enclosed between x = 2 and x = 4, so the limits of integration are from 2 to 4.
Finally, we can evaluate the integral to find the volume:
V = ∫(2π * (x + 5) * (x^(-4)))dx
To calculate this integral, you can use software or a calculator capable of symbolic integration or numeric approximation methods, such as numerical integration.