20.07.2022 - 21:24

A donuts is generated by rotating the circle, S, defined by (x-3)^2 +y^2 =4 about the y axis. If the management of Dunkin donuts wants to know how to price these donuts use integral calculus to help

Question:

A donuts is generated by rotating the circle, {eq}S {/eq}, defined by {eq}(x-3)^2 +y^2 =4 {/eq} about the y axis. If the management of Dunkin donuts wants to know how to price these donuts, use integral calculus to help them.

Set up a definite integral to determine the volume,{eq}V_1 {/eq}, of a donuts having no hole by rotating the right hand semi circle about the {eq}y {/eq} axis.

Answers (0)
  • Arlene
    April 7, 2023 в 02:03
    First, we need to find the equation of the right half of the circle by solving for y: {(x-3)^2 + y^2 = 4} becomes {(x-3)^2 = 4 - y^2} Taking the square root of both sides: {x - 3 = ±?(4 - y^2)} Simplifying: {x = 3 ±?(4 - y^2)} We only want the right half, which is {x = 3 + ?(4 - y^2)} Now, we can use the formula for the volume of a solid of revolution: {V = ? ?[a,b] f(x)^2 dx} where a and b are the limits of integration and f(x) is the function being rotated. In our case, f(x) is the right half of the circle, {f(x) = 3 + ?(4 - y^2)} Since we are rotating about the y-axis, the limits of integration are from -2 to 2 (the radius of the circle). Therefore, the definite integral to determine the volume of the donut with no hole is: {V1 = ? ?[-2,2] (3 + ?(4 - y^2))^2 dy } This integral can be evaluated using u-substitution or trigonometric substitution. Once we have calculated V1, we can use this volume and other factors like cost of ingredients and labor to determine the price of the donut.
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