2 The Volume of a Wine-glass We so far have been using triple integrals to compute the total moment of some function f over a region. E. But when this function f is just the constant 1, you end up with an integral of the form ll a and this just computes the volume of the region. E! This is very useful in practice, as the volumes of complicated regions can be calculated using inte- gration techniques. As an example here, imagine you are an industrial engineer working for a company that manufactures sets of high-end drinkware: wine glasses, cham- pagne flutes, and so on. The designers of the new collection have come up with a parametric model: a single model with several tune-able parameters they can set to produce different glasses with a uniform aesthetic. Their model describes the interior of the glass by a region Eng\" which can be tuned by changing two parameters a, b: Ea = {was 2) I w + yz) s z s a} The units here are centimeters, so the volume of the region is given in c1113, better known as milliliters. The design team speciw that wine glasses nmst be shorter and fatter, and so have b = (1 whereas champagne utes are tall and skinny, with l: = 3a. After doing a review of several cups across the industry, your manager finds the average volume of a red wine glass is 360mL. They then task you with taking the model from the design team, and constructing a wine glass as part of their collection that holds this same volume. What are the values of the parameters a, b that you need to report back to manufacturing, to have a glass of the right size? Hint: find the volume of Ea.b as a function of the parameters, by doing a triple integral in cylindrical coordinates. Use the equation given to find the bounds for z and the bounds for r! Then, after finding the general formula, use the fact that this is to be a wine glass to get Eat in terms of a single parameter, and then solve the equation for this volume to equal 360mL using a calculator or computer