Find the volume of the solid generated by rotating the region above 𝑦 = 12 and below 𝑦 = sin 𝑥 for 0≤𝑥≤𝜋 about the 𝑦-axis for 1 complete revolution.
the area ABOVE? y = 12???
That is infinite as is the area below y = sin x. There is no "between " I can see here.
Sorry typo. It should be 1/2
the curves intersect at 𝜋/6 and 5𝜋/6, so the volume, using shells, is
v = ∫[π/6,5π/6] 2πrh dx
where r=x and h=sinx - 1/2
v = ∫[π/6,5π/6] 2πx(sinx-1/2) dx = π^2(√3 - π/3)
using discs, it's tougher:
v = 2∫[1/2,1] π(R^2-r^2) dy
where R=π-arcsin(y) and r=arcsin(y)
v = ∫[1/2,1] π((π-arcsin(y))^2-(arcsin(y))^2) dy
= π^2(√3 - π/3)
To find the volume of the solid generated by rotating a region about the y-axis, we can use the method of cylindrical shells.
1. First, let's graph the region bounded by y = 12 and y = sin x for 0 ≤ x ≤ π to visualize the problem.
2. To find the limits of integration for the volume, we need to determine the x-values where the curves intersect.
At y = 12, the curve intersects at x = 0 and x = π.
At y = sin x, the curve intersects at x = 0, x = π/2, and x = π.
Since we want to find the volume for the complete revolution around the y-axis, our limits of integration are from x = 0 to x = π.
3. The radius of each cylindrical shell can be determined by finding the distance from the y-axis to the curve at a given x-value. In this case, the radius is simply x.
4. The height of each cylindrical shell can be determined by finding the difference in y-values between the two curves. In this case, the height is given by y = 12 - sin x.
5. The thickness of each cylindrical shell is infinitesimally small and can be represented by dx.
6. The volume of a cylindrical shell can be calculated using the formula: dV = 2πrh*dx, where r is the radius, h is the height, and dx is the thickness.
7. Integrating the volume expression over the limits of integration will give us the total volume.
Therefore, the volume of the solid generated by rotating the region above y = 12 and below y = sin x for 0 ≤ x ≤ π about the y-axis for 1 complete revolution is given by:
V = ∫[0 to π] 2πx(12 - sin x) dx