Find the volume of the solid obtained by rotating the region in the first quadrant enclosed by the curves y= tanx and y=2cosx between the bounds 0 and pi/2.
I need the setup.
The graph looks like this, and I see 2 different regions
in quadrant I bounded by the two curves.
www.wolframalpha.com/input/?i=plot+y+%3D+tanx+,+y+%3D+2cosx+from+0+to+%CF%80%2F2
- which one do you want
- you did not specify the line of rotation, e.g. the x-axis? the y-axis?
You will need their intersection
tanx = 2cosx
sinx/cosx = 2cosx
sinx = 2cos^2 x
sinx = 2(1 - sin^2 x)
sinx = 2 - 2sin^2 x
2sin^2 x + sinx - 2 = 0
sinx = (-1 ± √17)/4 = .78078 or -1.28... <---- not in quad I
This should get you going, once you decide what your rotation is
its around y=-1
then what do i do?
would it be
A= pi * ∫ 0 to 0.78 {(2cosx+1)^2 - (tanx+1)^2}
To find the volume of the solid, we will be using the method of cylindrical shells. Here's how we can set it up:
1. First, let's graph the two curves y = tan(x) and y = 2cos(x) in the first quadrant, between the bounds x = 0 and x = π/2.
2. The region enclosed by the curves in the first quadrant looks like a "banana" shape.
3. Next, imagine taking a thin vertical strip within the region and rotating it around the y-axis.
4. This rotation creates a cylindrical shell. The height of each shell is the difference in y-values between the two curves at a given x-value (the upper curve minus the lower curve).
5. The radius of each shell is the x-value itself.
6. To find the volume of each shell, we need the circumference, which is equal to 2π times the radius.
7. Finally, the volume of each shell is given by the product of the circumference and the height.
8. Repeat this process for all vertical strips within the region, from x = 0 to x = π/2.
9. Sum up the volumes of all the shells to find the total volume of the solid.
10. Use the appropriate integral to calculate this sum.
Let's call the volume V. The integral representation of this volume would be:
V = ∫[0,π/2] 2π * x * (2cos(x) - tan(x)) dx