MATH 550
Professor Girardi
Quiz 6
Due the week of 3/19

1. Evaluate the double integral:
   integral from y = 0 to y = 2     integral from x = y/2 to x = 1     sin(x²) dx dy  .
   Remark: Your answer should be an exact, and not an approximate, solution. You may not use a CAS on this problem.
2. Let R be a right pyramid with height h and a square base having sides of length s. Make a (rough) sketch of R and express the volume V of R as a triple integral. Then evaluate the triple integral to find the volume V.
   Hint: Place the square base in the xy-plane with the center at the origin and vertices on the x-axis & y-axis. Then, the words "right pyramid with height h" just means that the "tip" of the pyramid is at the point (0,0,h), ie, the line segment from the tip to the center of the base forms a "right angle" with the base and has length h.
3. Let D^* be the unit square in the uv-plane with vertices:
   1. A^* = (0,0)
   2. B^* = (0,1)
   3. C^* = (1,1)
   4. D^* = (1,0).
   Let D be the parallelogram in the xy-plane with vertices:
   1. A = ( 2,3)
   2. B = ( 3,7)
   3. C = (2,9)
   4. D = ( 1,5).
   Now:
   1. Make a rough sketch of D^* and D.
   2. Find a one-to-one and onto linear transformation T that takes
      1. A^* to A
      2. B^* to B
      3. C^* to C
      4. D^* to D .
      Your solution should be of the form: T ( u,v ) = (   x ( u,v )   ,   y ( u,v )  ) .
   3. What is the Jacobian Matrix J_T of T ?
   4. What is the determinate of J_T?
   5. Given your answer to part d and the fact that D^* has area 1, what is the area of D?
4. Let a, b, c > 0 and
   D = { (x , y , z )   :   (x / a) ^2   +   (y / b) ^2   +   (z / c) ^2   =   1 }
   D^* = { (u , v , w )   :   (u ) ^2   +   (v ) ^2   +   (w ) ^2   =   1 }   . So D is an ellipsoid and D^* is the unit sphere.
   1. Find a 1-to-1 transformation T so that T(D^* ) = D. So we want T(u , v , w ) = ( x (u , v , w )  ,   y (u , v , w )  ,   z (u , v , w )  )   . So we are looking for the functions: x = x (u , v , w )
      y = y (u , v , w )
      z = z (u , v , w ) . Hint: think of a sphere made of playdoo ... what would you do to the playdoo to transform the sphere into the ellipsoid?
   2. Express the volume of D as a triple integral over D.
   3. Express the volume of D as a triple integral over D^*.
   4. Find the volume of D ... you may use the fact that the unit sphere as volume 4pi/3.
      Hint: the volume of a sphere with radius r is (4pi/3) r^3 and if a=b=c, then the ellipsoid is a sphere with radius a. So if you answer in d. does NOT give (4pi/3) a^3 when you let a=b=c, then you know your answer is wrong.

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