Take the area enclosed by the curves y = sqrt(x), y = 1, and x = 4. Rotate it around the line x = 5. Find the volume.
While I understand the general problem would call for the shell method
∫[a,b] 2*pi*r*h dx
With r = 5 - x and
h = sqrt(x) - 1
But, what happens say if the question asked if it rotates around x = -6? Would my r equation be something like r = 6 + x?
My teacher always tends to show examples of rotation around the positive x axis, nothing negative yet so I was just curious.
Any help is greatly appreciated!
Yes, if you rotated about x = -6, r=x+6.
As you say, using shells of thickness dx,
v = ∫[1,4] 2π(5-x)(√x-1) dx = 103π/15
You can check your answer by using discs (washers) of thickness dy. In this case,
v = ∫[1,2] π(R^2-r^2) dy
where R=(5-x)=(5-y^2) and r=1
v = ∫[1,2] π((5-y^2)^2-1) dy = 103π/15
Thanks for the help Steve! Greatly appreciated!
Yes, you are correct that the shell method can be used to find the volume of the solid formed by rotating the area enclosed by the curves y = sqrt(x), y = 1, and x = 4 around the line x = 5.
The formula for the volume using the shell method is given by:
V = ∫[a,b] 2πrh dx
where r is the distance from the axis of rotation to the shell, and h is the height of the shell.
In this case, since you are rotating around the line x = 5, the distance from the axis of rotation to the shell can be expressed as r = 5 - x. Additionally, the height of the shell can be expressed as h = sqrt(x) - 1.
So the volume can be found by evaluating the integral:
V = ∫[a,b] 2π(5 - x)(sqrt(x) - 1) dx,
where a and b are the limits of integration, in this case a = 0 and b = 4.
Now, in the case where the rotation is around x = -6, the r equation would indeed be r = 6 + x, since the distance from the axis of rotation to the shell is now given by the signed distance from x = -6 to each point x on the curve. The height of the shell, h, would still be given by h = sqrt(x) - 1.
So the volume in this case would be found by evaluating the integral:
V = ∫[a,b] 2π(6 + x)(sqrt(x) - 1) dx,
with the appropriate limits of integration based on the given area enclosed by the curves.
I hope this explanation helps! Let me know if you have any further questions.