Exam 3 Math 850-007 Fall 04 Odenthal Name: Instructions: No books. No notes. Non-graphing calculators only. You are encouraged, although not required, to show your work.. Evaluate the iterated integral y 3y dx dy Mark your answer by placing a check in the appropriate box below. (a) (b) (c) 3 (d) 4 (e) 5 (f) 6 This is nothing but a computation. y 3y dx dy = = [3xy] x=y dx 6y 3y dx = [ 3y y 3] = ( 8) (3 ) =
Fall 04 Math 850 Exam 3 Page of. Find the limits of the iterated integral obtained from e x 0 y dx dy by changing the order of integration. Evaluate this new integral to find M. B D A C e x dy dx = M Mark your (five) answers by placing the appropriate letter (A, B, C, D, M) on the line left of the correct value. It is conceivable that more than one letter belongs on a single line. A C 0 B e / e M e / e + e + e/ e e x/ D x x y/ y y x e x 0 0 dy dx
Fall 04 Math 850 Exam 3 Page 3 of 3. Find the order of integration and limits of the (single) iterated integral that would, if evaluated, give the value of the indicated double integral where is the triangle in the xy-plane with vertices (0, 0), (, ), (0, ). y d = B D A C y E Mark your (five) answers by placing the appropriate letter (A, B, C, D, E) on the line left of the correct value. It is conceivable that more than one letter belongs on a single line. dx dy E dy dx A 0 B 3 C x y + x + y x y x y + x + y x y D x y After sketching the (triangular) region it s clear that we should integrate first with respect to y, or else we would have to break things into two iterated integrals. The relevant values for x are 0 x, and in this interval x y x. x da = x 0 x x dy dx
Fall 04 Math 850 Exam 3 Page 4 of 4. Find the order of integration and limits of the (single) iterated integral that would, if evaluated, give the value of the indicated double integral where is the region in the xy-plane bounded by the curves x = y(y ) and x = 3y(y ). B D y d = y E A C Mark your (five) answers by placing the appropriate letter (A, B, C, D, E) on the line left of the correct value. It is conceivable that more than one letter belongs on a single line. E dx dy dy dx 3 A 0 B 3 + 3 + x + + x + x + + x /3 + x /3 + x x + x /3 x /3 D y(y ) C 3y(y ) After sketching the (crescent shaped) region it s clear that we should integrate first with respect to x. The relevant values for y are 0 y, and in this interval 3y(y ) x y(y ). x d = y(y ) 0 3y(y ) x dx dy
Fall 04 Math 850 Exam 3 Page 5 of 5. Consider the double integral y d where is the region in the xy-plane inside the circle x + y = 4, to the right of the line x = 0, and above the line y =. Find the limits and integrand of the corresponding iterated integral in polar coordinates B D A C F drdθ Mark your (five) answers by placing the appropriate letter (A, B, C, D, F) on the line left of the correct value. It is conceivable that more than one letter belongs on a single line. 0 3 D 4 π π π / π /3 π /4 π /6 A π/6 π/4 π/3 B π/ π π sin θ r sin θ F r sin θ cos θ r cos θ r cos θ sec θ r sec θ r sec θ C csc θ r csc θ r csc θ π/4 π/6 csc θ r sin θ drdθ
Fall 04 Math 850 Exam 3 Page 6 of 6. Let be the region in the xy-plane that is bounded by the lines x y = ; x y = 4; x + y = 3; x + y = 8 and consider the function f(x, y) = x. Using the transformation u = x y and v = x + y find the function F = f(x(u, v), y(u, v)) and Jacobian J = the equality holds: f(x, y) dx dy = 8 4 3 F J du dv (x,y) (u,v) so that Mark your (two) answers by placing the appropriate letter (F or J) on the line left of the correct value. It is conceivable that more than one letter belongs on a single line. /5 /4 /3 J / 3 4 5 (u+v)/4 (u+v)/3 F (u+v)/ (u v)/4 (u v)/3 (u v)/ (v u)/4 (v u)/3 (v u)/ First observe that under the transformation the boundary lines of become the lines u = v = 3 u = 4 v = 8
Fall 04 Math 850 Exam 3 Page 7 of in the uv-plane. Next we calculate x = u + v and y = v u This allows us to find the Jacobian of the transformation (x, y) (u, v) = x u x v y u y v / / = = /. / / We then have x dx dy = = 8 4 3 8 4 3 u + v u + v 4 du dv du dv
Fall 04 Math 850 Exam 3 Page 8 of 7. Consider the solid defined by the inequalities 0 x ; 0 y ; and x z 4. The solid has density function δ(x, y, z) = y, mass M = 8 /3, and first moments M x =, M y = 6 /9, M z = 3 /5. Find the coordinates of the center of mass (x, y, z) of the solid. Mark your (three) answers by placing the appropriate letter (x, y, z) on the line left of the correct value. It is conceivable that more than one letter belongs on a single line. 0 /3 3 / y /3 3/ 3/6 6/3 x 3/4 4/3 5/3 3/5 5/ z /5 9/6 6/9 5/56 56/5 7/8 8/7 x = M x M = 8/3 = 3 4 y = M y M = 6/9 = 8/3 3 z = M z M = 3/5 = 8/3 5
Fall 04 Math 850 Exam 3 Page 9 of 8. Find the limits of the iterated integral using rectangular coordinates with the prescribed order of integration that would, if evaluated, give the value of the indicated triple integral where V is the solid in the first octant that lies inside the sphere x + y + z = 4 and above the plane z =. V x dv = (Same solid as problems 9 and 0.) B D F 0 C E x dz dy dx As you can see, one limit has been provided. Mark the remaining (five) answers by placing the appropriate letter (B, C, D, E, F) on the line left of the correct value. It is conceivable that more than one letter belongs on a single line. x y D 3 x 4 x 3 y 4 y C 0 E F 3 4 B x y 3 x y 4 x y 3 V = 3 3 x 0 0 4 x y x dz dy dx
Fall 04 Math 850 Exam 3 Page 0 of 9. Find the integrand and the limits of the iterated integral using cylindrical coordinates with the prescribed order of integration that would, if evaluated, give the value of the indicated triple integral where V is the solid in the first octant that lies inside the sphere x +y +z = 4 and above the plane z =. B D F x dv = G dz dr dθ (Same solid as problems 8 and 0.) V 0 0 E As you can see, two limits have been provided. Mark the remaining (five) answers by placing the appropriate letter (B, D, E, F, G) on the line left of the correct value. It is conceivable that more than one letter belongs on a single line. 0 π/6 E π/4 π/3 3 B π/ 4 π 3π/4 D F 3 4 r r r cos θ π 4 z z r sin θ G r cos θ r sin θ V = π/ 3 4 r 0 0 r cos θ dz dr dθ
Fall 04 Math 850 Exam 3 Page of 0. Find the integrand and the limits of the iterated integral using spherical coordinates with the prescribed order of integration that would, if evaluated, give the value of the indicated triple integral where V is the solid in the first octant that lies inside the sphere x +y +z = 4 and above the plane z =. B D F x dv = G dρ dφ dθ (Same solid as problems 8 and 9.) V 0 0 E As you can see, two limits have been provided. Mark the remaining (five) answers by placing the appropriate letter (B, D, E, F, G) on the line left of the correct value. It is conceivable that more than one letter belongs on a single line. 0 π/6 π/4 F D π/3 3 B π/ 4 π 3π/4 3 sin φ π cos φ E sec φ csc φ ρ sin φ sin θ ρ cos φ sin θ ρ sin φ ρ sin φ ρ sin φ cos θ ρ cos φ cos θ ρ cos φ ρ cos φ ρ 3 sin φ sin θ G ρ 3 sin φ cos θ ρ 3 cos φ sin θ ρ 3 cos φ cos θ
Fall 04 Math 850 Exam 3 Page of V = π/ π/3 0 0 sec φ ρ 3 sin φ cos θ dρ dφ dθ