Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.
y=1/x^4, y=0, x=2, x=9;
about y=–5
method: find the volume of the whole solid, then subtract the volume of the cylinder.
Volume of whole solid
= pi(integral)(1/x^4 + 5)^2 dx from 2 to 9
= pi[-1/(7x^4) - 10/(3x^3 + 25x] from 2 to 9
which came out to 551.077
the volume of the cylinder is
pi(5^2)(7) = 549.779
so the volume of the shape you described is 1.298
not too sure about my arithmetic.
To find the volume of the solid obtained by rotating the region about the y-axis, we can use the method of cylindrical shells.
First, we need to determine the limits of integration. In this case, the region is bounded by the curves y = 1/x^4, y = 0, x = 2, and x = 9. We can determine the limits of integration by solving the equations y = 1/x^4 and y = 0 for x.
Setting y = 1/x^4 and solving for x gives us x^4 = 1/y. Since we want to rotate the region about the y-axis, we're interested in the positive x-values for each y. We consider y > 0, which leads to x = (1/y)^(1/4).
Since x = 2 and x = 9 are the upper and lower limits of integration, respectively, we need to find the corresponding values of y. Substitute these x-values into the equation y = 1/x^4 to find the bounds for y: y = 1/2^4 = 1/16 and y = 1/9^4 = 1/6561.
Now that we have determined the limits of integration, we can set up the integral for finding the volume using cylindrical shells:
V = ∫(2π * r * h) dy,
where r is the distance from the axis of rotation to the shell (in this case, r = y + 5, since the axis of rotation is y = -5); h is the height of the shell (in this case, h = x2 - x1, which gives the difference in x-values for a given value of y); and dy is the differential height of the shell.
Therefore, the integral for finding the volume becomes:
V = ∫(2π * (y + 5) * (x2 - x1)) dy,
where x2 = (1/y)^(1/4) and x1 = 2.
V = ∫(2π * (y + 5) * ((1/y)^(1/4) - 2)) dy,
with the limits of integration being y = 1/16 and y = 1/6561.
Now, you can solve this integral to find the volume of the solid. Use a computer algebra system or integral-calculating software to evaluate the definite integral.