Final answer:
To find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis, you can use the method of cylindrical shells.
Step-by-step explanation:
To find the volume of the solid obtained by rotating the region bounded by the curves y= 1/x5,y=0,x=2, and x=5 about the axis y=−4, you can use the disk method.
The formula for the volume of the solid of revolution using the disk method is given by:
V=π∫ b/a [f(x)]2dx
where f(x) is the height of the function at a given x and π is the mathematical constant (approximately 3.14159).
In this case, the axis of rotation is y=−4, so you need to adjust the function f(x) accordingly. The height of the function will be f(x)+4.
So, the adjusted formula for the volume becomes:
V=π∫ b/a [f(x)+4]2dx
First, let's find the intersection points of y= 1/x5 and y=0 to determine the bounds of integration (a and b). Set y= 1/x5 equal to 0:
1/x5 =0
This equation has no solution since the numerator is always 1. Therefore, the region of interest is between x=2 and x=5.
Now, the integral becomes:
V=π∫5/2 [ 1/x5+4]2dx
You can simplify the integrand and then evaluate the integral. Keep in mind that this process involves some algebraic manipulation and integration steps.
Note: The integral might be complex to solve analytically. In such cases, numerical methods or software (like Mathematica, MATLAB, or Python with libraries like SciPy) can be used to approximate the result.