MATH 31B Midterm: MATH31B Midterm 2 2011 Spring Question

WeBWork ma215-001-umd-f17:HW11 16 6-8:10 Logged in as z n Math Kwebwork/ma215-001-umd-f17 / hw11 16 6-8/10 HW11 16 6-8: Problem 10 Previous Problem Problem List Next Problem (1 point) Write a triple integral including limits of integration that gives the volume of the cap of the solid sphere x2 + y2 2 20 cut off by the plane z = 2 and restricted to the first octant, an your integral, use theta, rho, and phi for θ, ρ and φ, as needed.) What coordinates are you using? (Enter cartesian, cylindrical, or spherical) With a = c= b= andf = Volume ss Note: You can earn partial credit on this problem. Preview My AnswersSubmit Answers

17.4. Lesson 17 Problem Set The correct answers are given for problems 1 and 2. Remember to check that the answer really is the maximum or minimum as required by the problem. This checking might take the form of evaluating the object function at critical points and end points of a closed interval of possible values, or if the interval in not closed, the check will probably involve investigating the sign of the first derivative. Warning: These problems can lead to headaches, chest pains, and baldness (from tearing out your hair). (1) Using the methods of this lesson, find two positive numbers with sum 100 and product as large as possible. (Answer: both numbers are 50.) (2) A tin can (right circular cylinder) with top and bottom is to have volume V. What dimensions (the radius of the bottom and the height) give the minimum total surface area? VI Answers: r -1/7 - 2T 2n square base and no top is to have n volume of 32 fts. What dinensions use the least amount of material (in other words what dimensions give minimum outside (3) A box with surface area)? (4) Find the largest area possible for a rectangle inscribed in a circle of radius r. (5) (Bonus question): Find the shortest (straight line) distance from the positive s-axis to the positive y-axis passing through the point (8. I).

The region is a sphere of radius 4. Find the limits of integration on the triple integral for the volume of the sphere using Cartesian, cylindrical, and spherical coordinates and the function to be integrated. For your answers θ = theta, φ = phi, and ρ = rho. Cartesian JA Jc Je where A-2 B=2 , c -sqrt(4-z/2) , D = sqrt(4-z 2) , E =|-sqrt(4-zh2 ) | , F = sqrt(4-z 2-y, and p(x, y, z) = 1 Cylindrical p(r, 0, z)dz dr do A JCJE where A 0 , B = 2pi D=2 Spherical where A-0 B=| pi ,D=2p. and p(p, θ, φ) = | e^2sin(theta) Note: You can earn partial credit on this problem. Preview My Answers Submit Answers