Page > of2 O zooM 4 MAT 273 Project, Spring 2024 Use Wolfram Alpha (or Symbolab) and Math3D.org (or Geogebra.org) to complete the following problems. If you are asked to explain something, type your answer into the Word document or upload a picture of the work to insert into the Word document. In many cases you will be asked to insert a graph or screenshot of work from the websites. You will submit the project as a PDF file on Canvas under Assignments. (You can save a Word document or a Google Sheets document as a PDF file.) 1. Let R be the 2D region in quadrant 1 bounded by the coordinate axes and the function y = 4 x%. Assume the density of the region is given by p(xy) =1+x+y. Write the integrals required to compute the center of mass of the region. Use a computer algebra system (symbolab or Wolfram Alpha) to evaluate the integrals. Take screenshots of the results. Then use the values obtained to write the coordinates in exact form. (25 pts.) 2. Let E be the 3D region in bounded below by z = x? + y? and above by z=8x*y?. Use math3d.org or Geogebra 3D to graph the region. Take a screenshot of it. Set up the integrals in cylindrical coordinates required to find the z-coordinate of the center of mass of the region if the density at any point is given by p(x,y,z2) =9 z. Use a CAS to evaluate the integrals and find the z-coordinate. Take screenshots of the results. (30 pts.) 3. Suppose you want to find the absolute extrema of f(x,y) = xy* subject to the constraint y? +4xy = 108. a. Use gradients to write the system of equations that must be solved in order to find critical points. Write or type the equations below. b. Use Wolfram Alpha (not Symbolab, as it doesn't do this well) to solve the system from (a). This uses a simple \"solve\" command, followed by the equations, separated by commas. You may use k or some other symbol in the place of lambda if you prefer, although the lambda symbol is available on the math toolbar. Paste a screenshot of the result below. c. Find the function values at the critical points and determine which gives the absolute minimum and which the absolute maximum. (20 pts.) TR vl O zooM 4 MAT 273 Project, Spring 2024 4. Use the function f(xy) = %x' 4x y? 2x? +8y? to carry out the following: (25 pts.) a) Graph the function on the rectangular domain 3