Math 52 First Midterm January 29, 2009

Transcription

1 Math 5 First Midterm Januar 9, 9 Name : KEY Section Leader: Josh Lan Xiannan (Circle one) Genauer Huang Li Section Time: : : :5 :5 (Circle one) This is a closed-book, closed-notes eam. No calculators or other electronic devices will be permitted. You have hours. If ou finish earl, ou must hand our eam paper to a member of teaching staff. In order to receive full credit, please show all of our work and justif our answer. You do not need to simplif our answers unless specificall instructed to do so. If ou need etra room, use the back sides of each page. There is alsoablankpageatthe end of the eam for ou to use. Do not unstaple or detach pages from this eam. Please sign the following: On m honor, I have neither given nor received an aid on this eamination. I have furthermore abided b all other aspects of the honor code with respect to this eamination. Signature: Total

2 Math 5, Winter 9 First Midterm Januar 9, 9 Page of 9. ( points) Sketch the region of integration R in the -plane for the integral below. Then reverse the order of integration; i.e., write an iterated integral with eplicit limits in which the - integration is done first. NO EVALUATION OF INTEGRALS IS REQUIRED. R g(, )dd = 5 5 g(, ) dd =5 (, ) = = = 5 =5 5 g(, ) d d

3 Math 5, Winter 9 First Midterm Januar 9, 9 Page of 9. (8 points) Below is an integral over a solid region in -space. Rewrite it as an iterated integral in spherical co-ordinates over the same region. Evaluate the integral which is written in spherical co-ordinates. 4 ( + ) ddd = 4 ( + ) = shadow + r cos φ 4ρ sin φ ρ ( dρ dφ dθ =4 4 =8 +4 =4. cos φ ) sin φ dφ u du ( ) u =cosφ

4 Math 5, Winter 9 First Midterm Januar 9, 9 Page 4 of 9. ( points) Consider the circle which is described in polar coordinates b r =sinθ. Let D be a plate consisting of points inside this circle and outside the circle r =. Supposethedensitδ(, ) at a point (, ) equals. a) Set up a double integral in polar co-ordinates, with eplicit limits of integration, for the mass of the plate. DO NOT EVALUATE THE INTEGRAL. answer = 5 sinθ r sin θ dr dθ r =sinθ 5 (, ) r = Note: r =sinθ + = +( ) = b) Set up a formula for the -coordinate ȳ of the center of mass of the plate in terms of integrals in polar coordinates with eplicit limits of integration. DO NOT COMPUTE the value for ȳ. Eplainwhouknow,withoutcomputation,thatthe-coordinate for the center of mass is. Note that both the region and the densit function are smmetric across the -ais. Consequentl =. ȳ = 5 5 sinθ r sin θ dr dθ sinθ r sin θ dr dθ

5 Math 5, Winter 9 First Midterm Januar 9, 9 Page 5 of 9 4. ( points) Consider the cone in R determined b the condition that the spherical coordinate φ (angle with positive -ais) satisfies φ. Let W be the portion of this cone which lies between the sphere of radius and the sphere of radius. a) Compute the volume of W. volume = ρ sin φ dρ dφ dθ ( = ) sin φ dφ = 7 b) Find the average distance of a point in W to the origin. average distance = 7 = ( ) 4 = 45 8 ρ sin φ dρ dφ dθ

6 Math 5, Winter 9 First Midterm Januar 9, 9 Page of 9 5. (8 points) Consider the triangular region D in the (, )-plane bounded b the lines =, =,and =. LetW be the -dimensional solid obtained b rotating D all the wa around the -ais. (W is like a donut with a triangular cross-section.) Set up an iterated integral in clindrical coordinates that computes the volume of W and then evaluate that integral. = = (, ) (, ) (, ) = r rddrdθ = r rdr ( r ) = r = 5.

7 Math 5, Winter 9 First Midterm Januar 9, 9 Page 7 of 9. ( points) Let D be the diamond-shaped region in the plane with vertices at (, ), (, ), (, ), (, ). Consider the transformation T from the uv plane to the plane defined b T (u, v) =( u v, u + v )=(, ). a) Draw a sketch of D in the plane and find the rectangle C in the uv plane which is taken b T to D. v T (, ) (, ) C (, ) (, ) D (, ) (, ) u b) Set up an integral (using the transformation T as a change of coordinates) over the region C in the uv-plane that can be used to compute the following integral over D: ( ) sin ( + ) dd. Then COMPUTE the integral over C. Observe, Thus D D ( ) = u v ( + ) = u v (, ) (u, v) = ( ) sin ( + ) d d = = = 4 C u + v + u + v = v = u =. ( v) (sin u) (, ) (u, v) du dv v sin ududv sin (u)du = 44 use sin u = cos(u)

8 Math 5, Winter 9 First Midterm Januar 9, 9 Page 8 of 9 7. (8 points) a) Set up an iterated integral that computes the integral of f(,, ) = over the following region in R :,, +, +. Integrate first in the -direction and then the other two directions in either order. Do NOT evaluate the integral. = + = and = = and = = + = shadow = + = = + ddd b) Sketch the region in part a). Set up the limits of integration in order to repeat part a), this time integrating first in the -direction. Do NOT compute the integral(s). = and = = + = and = = shadow (, ) (, ) = = (, ) + = = ddd+ ddd

9 Math 5, Winter 9 First Midterm Januar 9, 9 Page 9 of 9 Scratch Paper