The bean counting method that was described in lecture (and in Chapter 5 of the course reader) for determining a is known as Monte Carlo integration." It is useful for integrating complicated functions, including cases where it may not even be clear how to properly express the integral. For example, the expression Mr.w.a) = =0 (1-7--)*+ ). 3 y? 22 + x2 23 + h(x, y, z) = 320 (1) can be used to generate a three-dimensional shell" (i.e., a level surface) by setting h(x, Y, z) = 0. You will investigate the utility of Monte Carlo integration for determining the volume contained within the shell. (a) Briefly describe a problem you might encounter when trying to formulate an integral to de- termine the shell volume. (b) In MATLAB, plot the shell (hint: this is an 'implicit' expression) and use the plot to determine an appropriately sized cube (centered on the origin) that entirely contains the shell. Use the plot option 'daspect([1 1 1])'. Take the total cube edge length to be equal to the smallest integer value that contains the shell. What is the length? (c) In terms of the variable h, write an expression for the region inside the shell (not including the shell itself). (d) In MATLAB, estimate the volume within the shell using Monte Carlo integration. Assume the "falling" beans "land" based on uniform distributions. Generate a convergence plot: plot the volume estimate as a function of the number of beans used for the estimate. Use 100, 000 beans to generate your results, but create one plot point every hundred beans (i.e., 1,000 plot points total). Display the resulting volume estimate. The bean counting method that was described in lecture (and in Chapter 5 of the course reader) for determining a is known as Monte Carlo integration." It is useful for integrating complicated functions, including cases where it may not even be clear how to properly express the integral. For example, the expression Mr.w.a) = =0 (1-7--)*+ ). 3 y? 22 + x2 23 + h(x, y, z) = 320 (1) can be used to generate a three-dimensional shell" (i.e., a level surface) by setting h(x, Y, z) = 0. You will investigate the utility of Monte Carlo integration for determining the volume contained within the shell. (a) Briefly describe a problem you might encounter when trying to formulate an integral to de- termine the shell volume. (b) In MATLAB, plot the shell (hint: this is an 'implicit' expression) and use the plot to determine an appropriately sized cube (centered on the origin) that entirely contains the shell. Use the plot option 'daspect([1 1 1])'. Take the total cube edge length to be equal to the smallest integer value that contains the shell. What is the length? (c) In terms of the variable h, write an expression for the region inside the shell (not including the shell itself). (d) In MATLAB, estimate the volume within the shell using Monte Carlo integration. Assume the "falling" beans "land" based on uniform distributions. Generate a convergence plot: plot the volume estimate as a function of the number of beans used for the estimate. Use 100, 000 beans to generate your results, but create one plot point every hundred beans (i.e., 1,000 plot points total). Display the resulting volume estimate