A donut is generated by rotating the circle, S, defined by (x-3)^2 +y^2 =4 about the y axis. If the anagement of dunkin donuts wants to know how to price these donuts use intergral calculus to help them
(i) set up a definite intergral to determine the volume,V1, of a donut having no hole by rotating the right hand semi circle about the y axis.
Please help!
Bogus problem!
The right semicircle is 3 units from the y axis. If you rotate that piece around the y-axis, it will have a 6-unit diameter hole in the middle.
However, the integral as described above can take advantage of the symmetry about the x-axis, and is thus found using washers:
2∫[0,2] π (R^2-r^2) dy
where R = x = 3+√(4-y^2) and r=3
= 2π ∫[0,2] (3+√(4-y^2))^2 - 9 dy
= 4π (3π + 8/3)
Or, integrating over x using shells,
2∫[3,5] 2πrh dx
where r = x and h = y = √(4-(x-3)^2)
= 4π ∫[3,5] x√(4-(x-3)^2) dx
= 4π(3π + 8/3)
Or, using the Theorem of Pappus,
area of semi-circle is 1/2 π * 2^2 = 2π
The centroid is at 3+(4r/3π) = 3+(8/3π)
So, the volume is
2π(2π*(3+(8/3π)))
= 2π(6π + 16/3)
= 4π(3π + 8/3)
To set up the definite integral to determine the volume, V1, of a donut with no hole, we need to consider the method of cylindrical shells.
The first step is to express the equation (x-3)^2 + y^2 = 4 in terms of y. We can rewrite it as:
(x - 3)^2 = 4 - y^2
x - 3 = ±√(4 - y^2)
x = 3 ± √(4 - y^2)
Now, we need to consider the range of y-values over which the circle S is defined. Since the equation represents a circle with a radius of 2 centered at (3, 0), it will be defined over the interval [-2, 2] (from -2 to 2 inclusive).
To find the volume V1, we need to rotate the right-hand semicircle around the y-axis. This generates a donut shape with no hole, which can be thought of as a solid cylinder with a missing cylindrical region.
To calculate the volume, we divide the problem into infinitesimally thin cylindrical shells. Each shell has a height dy and a radius x, which is equal to 3 + √(4 - y^2).
The volume of each cylindrical shell is given by 2πx * dy (since it accounts for both sides of the donut). Therefore, the integral to determine the volume V1 is:
V1 = ∫[a,b] 2πx * dy
Here, the limits of integration, [a,b], correspond to the range of y-values over which the circle S is defined. In this case, a = -2 and b = 2.
So, the definite integral to determine the volume V1 becomes:
V1 = ∫[-2,2] 2π(3 + √(4 - y^2)) * dy
Once the integral is evaluated, it will provide the volume of the donut with no hole. The management of Dunkin' Donuts can consider factors like the cost of ingredients, overhead costs, and profit margin to determine the pricing for these donuts.