MAT 240 Maple Lab 4 Optimization and Iterated Integrals

For the following problems you can use the code given in class in order to answer the following problems from the book. Exercise 1 [10 pts]: Create contour diagrams for functions 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 for k = -2, -1, 0, 1, 2. Use these figures to classify the critical point at (0,0) for each value of k. Explain your observations using the discriminant. Exercise 2 [10 pts]: Let 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 in the region R where x, y > 0. Explain why f must have a global minimum at some point in R and find the global minimum. Exercise 3 [10 pts]: Use the method of Lagrange multipliers to find the maximum and minimum values of f subject to the given constraint, if such values exist. If the values do not exist, give justification. 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

(a) Plot 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 for R = {(x,y) | 0< x < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjA7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy /2, 0< y < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjA7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy /4 }. Find the exact volume using an iterated integral. restart: with(plots): with(student); plot3d(sin(x+y),x=0..Pi/2,y=0..Pi/4,axes=boxed); V:=Int(Int(sin(x+y),x=0..Pi/2),y=0..Pi/4): V=value(V); (b) Approximate the above integral with left and right Riemann double sums. Use m = 6 and n = 10 subdivisions. Vleft:=leftsum(leftsum(sin(x+y),x=0..Pi/2,6),y=0..Pi/4,10): Vleft=evalf(Vleft); Vright:=rightsum(rightsum(sin(x+y),x=0..Pi/2,6),y=0..Pi/4,10): Vright=evalf(Vright);

Find the volume of 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 for R = {(x,y) | 0< x < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjA7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy /2, 0< y < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjA7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy /4 }using the Maple function Doubleint. V:=Doubleint(sin(x+y),x=0..Pi/2,y=0..Pi/4): V=value(V);

Consider the solid between the paraboloids 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 and 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 , and inside the cylinder 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 . a) Plot the paraboloids on the appropriate domain. f:=2*x^2+y^2; g:=8-x^2-2*y^2; plot3d({f,g},x=-1..1,y=-sqrt(1-x^2)..sqrt(1-x^2),axes=normal); NOTE: The domain of integration is the circle 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 which can be described by 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 and 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 ; . Also note that for x and y inside this domain the paraboloid g is above f. We can plot the surfaces and the domain of integration using the following commands: with(plottools): q:=plot3d({f,g},x=-1..1,y=-sqrt(1-x^2)..sqrt(1-x^2),axes=boxed): p:=implicitplot(x^2+y^2=1,x=-1..1,y=-1..1,thickness=4): h:=transform((x,y)->[x,y,0]): display({q,h(p)}); b) Find the volume of the solid using a Double Integral. V:=Doubleint(g-f,y=-sqrt(1-x^2)..sqrt(1-x^2),x=-1..1): V=value(V);

Just as plot3d can plot a surface of the form z = f(x,y) you can tell Maple to use cylindrical coordinates, and it will assume you have the surface expressed r as a function of z and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2J1EnJiM5NTI7RicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGNy9GNlEnbm9ybWFsRic= . (NOTE: MAPLE will actually express it the other way than we are used to it: r = z. (not z = r).

Lets look at the top half of the cone 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 . In cylindrical coordinates, this would be z = r. plot3d(z, theta=0..2*Pi, z=0..4,axes=normal,coords=cylindrical); IMPORTANT: The order in which you set the range for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2J1EnJiM5NTI7RicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGNy9GNlEnbm9ybWFsRic= and z when plotting in cylindrical coordinates is very important! Maple expects the range for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2J1EnJiM5NTI7RicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGNy9GNlEnbm9ybWFsRic= first and for z second. Volume: We can find the volume of the cone for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYnUSJ6RicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR0Y2LyUsbWF0aHZhcmlhbnRHUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJy1JI21vR0YkNi1RJSZsZTtGJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkcvJSlzdHJldGNoeUdGRy8lKnN5bW1ldHJpY0dGRy8lKGxhcmdlb3BHRkcvJS5tb3ZhYmxlbGltaXRzR0ZHLyUnYWNjZW50R0ZHLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVi1JI21uR0YkNiZRIjRGJ0Y0L0Y6Rj5GPEZDRitGQw== in cylindrical coordinates. Note that the projection of the cone in the xy plane is given by the circle 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 so that the limits for r are 0 and 4 while z varies from r (the equation of the cone) to 4. V:=Int(Int(Int(r,z=r..4),r=0..4),theta=0..2*Pi); value(%); If we set up the integral that gives the volume in rectangular coordinates we have much more complicated limits. Indeed the integral is so complicated that Maple cannot give an exact answer. To evaluate the integral we need to use evalf which gives a numerical approximation. Int(Int(Int(1,z=sqrt(x^2+y^2)..4),y=-sqrt(16-x^2)..sqrt(16-x^2)),x=-4..4); value(%); evalf(%);

Lets look at the paraboloid 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 . This gives 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 , which we must express in terms of r, so LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYnUSJyRicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR0Y2LyUsbWF0aHZhcmlhbnRHUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkcvJSlzdHJldGNoeUdGRy8lKnN5bW1ldHJpY0dGRy8lKGxhcmdlb3BHRkcvJS5tb3ZhYmxlbGltaXRzR0ZHLyUnYWNjZW50R0ZHLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVi1JJm1zcXJ0R0YkNiMtRiw2J1EiekYnRjRGN0Y5RjxGK0ZDRitGQw== . I have put in a grid option so you can see how it affects the plot. Experiment with the numbers in the grid specification. plot3d(sqrt(z),theta=0..2*Pi,z=0..4,axes=normal, coords=cylindrical,grid=[36,25]); In these plots, Maple knows the definitions of the r and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2J1EnJiM5NTI7RicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGNy9GNlEnbm9ybWFsRic= variables because of the coords specification. There is another way to plot these surfaces. PLEASE NOTE THIS: we can use the command plot3d by giving expressions for the x, y, and z coordinates. The cylindrical coordinate transformation is given by 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 . Using this method, which is called plotting surfaces parametrically, we do not use the cylindrical coordinate specification because we are plotting the x, y, and z coordinates. We can plot the cone parametrically by using 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 plot3d([r*cos(theta),r*sin(theta),r],r=0..4,theta=0..2*Pi,axes=normal); Similarly, we could use the more familar expression 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 when plotting the paraboloid: plot3d([r*cos(theta),r*sin(theta),r^2],r=0..4,theta=0..2*Pi, axes=normal);

Plot the sphere with radius 5 in spherical coordinates. Maple assumes you have your surface expressed as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy as a function of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy when you use spherical coordinates, and it expects you to list the ranges for theta first and phi second. plot3d(5,theta=0..2*Pi,phi=0..Pi,coords=spherical,scaling=constrained,axes=boxed); To plot the surface parametrically we can use 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 plot3d([5*cos(theta)*sin(phi),5*sin(theta)*sin(phi),5*cos(phi)],theta=0..2*Pi,phi=0..Pi,axes=boxed); The volume of the sphere can be computed by the following triple integral (note that the integrand function is given by 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 : V:=Int(Int(Int(rho^2*sin(phi),rho=0..5),theta=0..2*Pi),phi=0..Pi); value(%);

Graph the solid bounded by the plane x + y + z =1 and the paraboloid 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 and find its exact volume. Solution: We first graph the two functions: f:=4-x^2-y^2; f := 4-x^2-y^2; g:=1-x-y; plot3d({f,g},x=-2.5..2.5,y=-2.5..2.5,axes=boxed); The domain of integration D is the projection on the xy plane of the curve of intersection of the two surfaces. To find this projection we need to eliminate the z from the two equations. This gives 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 or 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 . This is the equation of a circle (You can graph it using implicitplot to confirm that it is a circle or you can complete the square which gives 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 . In order to write the double integral we need to express either x as a function of y or y as a function of x. After completing the square this sould be fairly easy but we can have Maple do the work for us: funct:=solve(f=g,y); f1:=funct[2]; f2:=funct[1]; Of course we get two functions: one for the lower bound of the circle and one for the upper bound. We also need to find the x- values corresponding to the intersections of these two curves xlimit:=solve(f1=f2,x); x1:=xlimit[1]; x2:=xlimit[2]; To check that our findings are correct we plot the two surface together with the functions that we found as lower and upper bound of the domain of integration. We also plot the curve of intersection of the two surfaces. The parametric equations of this curve of intersection were derived by parametrizing the circle and then substituting the expressions for x and y in the equation of the plane (or the paraboloid). with(plottools): q:=plot3d({f,g},x=-2.5..2.5,y=-2.5..2.5,axes=boxed): p:=plot({f1,f2},x=x1..x2,thickness=4,color=red): r:=spacecurve([1/2+sqrt(7/2)*cos(t),1/2+sqrt(7/2)*sin(t),-sqrt(7/2)*(cos(t)+sin(t))],t=0..2*Pi,color=black,thickness=2): h:=transform((x,y)->[x,y,0]): display({q,r,h(p)}); From the graph above it looks like the red circle is indeed the projection on the xy plane of the intersection of the two curves. If we want another check we can plot the two surfaces with domain the circle. plot3d({f,g},x=x1 ..x2, y = f1..f2,axes=boxed); We get indeed the solid bounded above by the paraboloid and below by the plane Now we can evaluate the volume of the solid. V:=Doubleint(f-g,y=f1..f2,x=x1..x2): V=value(V);

Exercise 1 [15 pts]: (a) Plot f(x,y) = y sin(x) for R = {(x,y) | LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNiYtSSNtb0dGJDYtUSIoRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSV0cnVlRicvJSpzZXBhcmF0b3JHUSZmYWxzZUYnLyUpc3RyZXRjaHlHRjwvJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRLDAuMTY2NjY2N2VtRicvJSdyc3BhY2VHRkwtRiM2Ji1JI21uR0YkNiRRIjBGJ0Y3LUY0Ni1RJSZsZTtGJ0Y3L0Y7Rj9GPS9GQUY/RkJGREZGRkgvRktRJjAuMGVtRicvRk5GZW4tRiw2JVEieEYnLyUnaXRhbGljR0Y8L0Y4USdpdGFsaWNGJ0Y3LUY0Ni1RIilGJ0Y3RjpGPUZARkJGREZGRkhGSi9GTlEsMC4xMTExMTExZW1GJ0Y3RlUtRiw2JVEnJiM5NjA7RicvRltvRj9GN0Y3RitGNw== , 0 < y < 1 }. (b) Approximate the above integral with left and right Riemann double sums. Use m = 5 and n = 15 subdivisions. (c) Find the volume using iterated or double integral. Exercise 2 [15 pts]: a) For LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNictRiw2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RMCZBcHBseUZ1bmN0aW9uO0YnLyUlc2l6ZUdRIzE0RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZHLyUpc3RyZXRjaHlHRkcvJSpzeW1tZXRyaWNHRkcvJShsYXJnZW9wR0ZHLyUubW92YWJsZWxpbWl0c0dGRy8lJ2FjY2VudEdGRy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlYtRiM2Ji1GPTYtUSIoRidGQy9GRkY4RkgvRktGOEZMRk5GUEZSL0ZVUSwwLjE2NjY2NjdlbUYnL0ZYRltvLUYjNiYtRiw2JVEieEYnRjZGOS1GPTYtUSIsRidGQ0ZFL0ZJRjhGSkZMRk5GUEZSRlQvRlhRLDAuMzMzMzMzM2VtRictRiw2JVEieUYnRjZGOUZDLUY9Ni1RIilGJ0ZDRmhuRkhGaW5GTEZORlBGUkZqbi9GWFEsMC4xMTExMTExZW1GJ0ZDRitGQy1GPTYtUSI9RidGQ0ZFRkhGSkZMRk5GUEZSL0ZVUSwwLjI3Nzc3NzhlbUYnL0ZYRmRwLUYjNiZGKy1JJW1zdXBHRiQ2JS1GLDYlUSN4eUYnRjZGOS1JI21uR0YkNiRRIjJGJ0ZDLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YrRkNGK0ZDRitGQw== over R = {(x , y) | 0 < x < 1, 2 x < y < x + 1 }, plot the region over which the integration takes place (in the xy plane) , and plot the surface with the domain of the region of integration. b) evaluate the integral. c) Write an equivalent integral(s) with the order of integration reversed and evaluate again. Observe that the result is the same.

Exercise 1 [5 pts]: Cylindrical coordinates Plot the sphere 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 using CYLINDRICAL coordinates. (remember to write the limits for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy first). Exercise 2 [10 pts]: Spherical coordinates a) Plot the lower half of the sphere LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY3LUkjbWlHRiQ2I1EhRictRiM2OEYrLUYjNjtGKy1GIzY3RistSSVtc3VwR0YkNiUtRiw2NlEieEYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTRGJy8lJWJvbGRHUSZmYWxzZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUqdW5kZXJsaW5lR0ZDLyUqc3Vic2NyaXB0R0ZDLyUsc3VwZXJzY3JpcHRHRkMvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lJ29wYXF1ZUdGQy8lK2V4ZWN1dGFibGVHRkMvJSlyZWFkb25seUdGQy8lKWNvbXBvc2VkR0ZDLyUqY29udmVydGVkR0ZDLyUraW1zZWxlY3RlZEdGQy8lLHBsYWNlaG9sZGVyR0ZDLyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR0ZDLyUwZm9udF9zdHlsZV9uYW1lR1ErX2NzdHlsZTI3NEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNjZRIjJGJ0Y7Rj5GQS9GRUZDRkdGSUZLRk1GUEZTRlVGV0ZZRmVuRmduRmluRltvRl1vL0Zhb1Enbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGOy9GP1EjMTJGJ0ZBRmdvRkdGSUZLRk1GUEZTRlVGV0ZZRmVuRmduRmluRltvRmhvLUkjbW9HRiQ2P1EiK0YnRjtGPkZBRmdvRkdGSUZLRk1GUEZTRlVGV0ZZRmVuRmduRmluRltvRl1vRmhvLyUmZmVuY2VHRkMvJSpzZXBhcmF0b3JHRkMvJSlzdHJldGNoeUdGQy8lKnN5bW1ldHJpY0dGQy8lKGxhcmdlb3BHRkMvJS5tb3ZhYmxlbGltaXRzR0ZDLyUnYWNjZW50R0ZDLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGY3EtRiM2N0YrLUY2NiUtRiw2NlEieUYnRjtGPkZBRkRGR0ZJRktGTUZQRlNGVUZXRllGZW5GZ25GaW5GW29GXW9GYG9GY29Gam9GK0Y7Rl1wRkFGZ29GR0ZJRktGTUZQRlNGVUZXRllGZW5GZ25GaW5GW29GaG9GX3AtRiM2N0YrLUY2NiUtRiw2NlEiekYnRjtGPkZBRkRGR0ZJRktGTUZQRlNGVUZXRllGZW5GZ25GaW5GW29GXW9GYG9GY29Gam9GK0Y7Rl1wRkFGZ29GR0ZJRktGTUZQRlNGVUZXRllGZW5GZ25GaW5GW29GaG9GK0Y7Rl1wRkFGZ29GR0ZJRktGTUZQRlNGVUZXRllGZW5GZ25GaW5GW29GaG8tRmBwNj9RIj1GJ0Y7Rj5GQUZnb0ZHRklGS0ZNRlBGU0ZVRldGWUZlbkZnbkZpbkZbb0Zdb0Zob0ZjcEZlcEZncEZpcEZbcUZdcUZfcS9GYnFRLDAuMjc3Nzc3OGVtRicvRmVxRmhyLUZkbzY2USI0RidGO0Y+RkFGZ29GR0ZJRktGTUZQRlNGVUZXRllGZW5GZ25GaW5GW29GXW9GaG9GO0ZdcEZBRmdvRkdGSUZLRk1GUEZTRlVGV0ZZRmVuRmduRmluRltvRmhvRitGO0ZdcEZBRmdvRkdGSUZLRk1GUEZTRlVGV0ZZRmVuRmduRmluRltvRmhv using SPHERICAL COORDINATES (Hint: remember to set LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy correctly and remember to set the range for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy first) b) Find the volume of the lower half of the sphere by setting up a triple integral in spherical coordinates.