Problem 4: [40 pts] Donuts! A reasonable way to model a. donut is by revolving a region R, whose bmmdaty is a circle, about the yaxis. A rendition ofthe donut The region R in the Icyplane Some actual measurements we can take from the donut that we can use for the modeling include the following. o The distance a. from the center of the hole in the donut to the inner ring. 0 The distance I) from the center of the hole in the donut to the outer ring. A. [3 pts] Find an equation for the circle that forms the boundary of R. Leave a. and b in your nal answer. B. [3 pts] Often times, the measurements that we can reasonably nd lead to notationally cumbersome equations in our modeling or we can note that certain quantities appear frequently in the model. In these instances, it is often helpful to introduce auxiliary variables to make the notation look cleaner}. a+b bo Seth=Tandr= 2 , and deduce that the equation of the circle in Part A becomes (1' 3:): +3;2 2 r . (Problem 4 continued) C. [10 pts] Use the washer method to show that the volume of the donut is given by the integral below. In order to receive full credit, you must draw an appropriate horizontal or vertical slice on the image below that, when revolved, forms a washer and label the inner and outer radii. D. [5 pts] Simplify the volume integral above. Then, explain how symmetry can be used to conclude that V = 8wh\\/72 - yz dy