Consider the given curves to do the following.
64 y = x^3, y = 0, x = 4
Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about y = 1.
shells are just cylinders, so
v = ∫[0,1] 2πrh dy
where r = 1-y and h = 4-x = 4-4∛y
v = 8π∫[0,1] (1-y)(1-∛y) dy
= 8π∫[0,1] 1 - y^(1/3) - y + y^(4/3) dy
= 8π (y - 3/4 y^(3/4) - 1/2 y^2 + 3/7 y^(3/7) [0,1]
= 8π (1 - 3/4 - 1/2 + 3/7)
= 8π (5/28)
= 10π/7
Just for a check, try using discs (washers)
Now, we have
v = ∫[0,4] π(R^2-r^2) dx
where R=1 and r = 1-y = 1-x^3/64
v = π∫[0,4] (1-(1-x^3/64)^2) dx
= π∫[0,4] (x^3/32 - x^6/4096) dx
= π (x^4/128 - x^7/26872) [0,4]
= 10π/7
To find the volume generated by rotating the region bounded by the curves 64y = x^3, y = 0, and x = 4 about y = 1 using the method of cylindrical shells, follow these steps:
1. Determine the limits of integration:
- The region is bounded by y = 0 and y = 1, so the limits of integration for y will be from 0 to 1.
2. Find the height of the cylindrical shell:
- The height of each cylindrical shell will be given by the difference between the y-coordinate of the curve 64y = x^3 and the y-coordinate of the axis of rotation, y = 1. So the height for each shell will be 1 - y.
3. Calculate the radius of the cylindrical shell:
- The radius of each cylindrical shell will be given by the x-coordinate of the curve 64y = x^3. Rearranging the equation, we have x = (64y)^(1/3).
4. Express the volume element:
- The volume element of each cylindrical shell will be equal to the height multiplied by the circumference of the shell (2πr). So, V_element = 2πr * height.
5. Set up and evaluate the integral:
- The integral to find the total volume is given by V = ∫(V_element) dy, where the limits of integration are from 0 to 1.
V = ∫[0→1] 2π((64y)^(1/3)) * (1 - y) dy
6. Integrate the integral:
- Integrate the function 2π((64y)^(1/3)) * (1 - y) with respect to y over the limits 0 to 1 to find the volume:
V = ∫[0→1] 2π((64y)^(1/3)) * (1 - y) dy
7. Evaluate the integral:
- Calculate the definite integral using the antiderivative and substitute the limits of integration:
V = 2π * ∫[0→1] ((64y)^(1/3) - (64y)^(4/3)) dy
8. Calculate the integral:
- Integrate the function ((64y)^(1/3) - (64y)^(4/3)) with respect to y over the limits 0 to 1 and multiply by 2π to find the volume.
To find the volume V generated by rotating the region bounded by the given curves about y = 1 using the method of cylindrical shells, you can follow these steps:
1. First, sketch the region bounded by the curves. In this case, you have the curve 64y = x^3, the x-axis (y = 0), and the line x = 4.
2. Determine the limits of integration. Since the region is bounded by y = 0 and y = 1, the limits of integration for y will be from 0 to 1.
3. Express the equation of the curve 64y = x^3 in terms of y. By rearranging, you get x = (64y)^(1/3).
4. Determine the height of each cylindrical shell. In this case, the height of each shell will be 1 minus the y-coordinate of the shell, since the region is being rotated about y = 1.
5. Express the radius of each cylindrical shell. The radius of each shell will be the x-coordinate of the curve at a given value of y, which is x = (64y)^(1/3).
6. Express the differential element of volume for each cylindrical shell. The differential element of volume is given by dV = 2πrh dy, where r is the radius of the shell and h is its height.
7. Integrate the differential element of volume from y = 0 to y = 1 to find the total volume V. The integral becomes V = ∫(0 to 1) 2π(64y)^(1/3)(1-y)dy.
8. Evaluate the integral to find the volume V.
By following these steps, you should be able to calculate the volume V generated by rotating the region bounded by the given curves about y = 1 using the method of cylindrical shells.