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Unformatted text preview: APPM 2350 EXAM 3 FALL 2010 INSTRUCTIONS: Books, notes, crib sheets, and electronic devices are not permitted. Write your (1) name, (2) instructor’s name, and (3) recitation number on the front of your bluebook. Work all problems. Show and explain your work clearly. Note that a correct answer with incorrect or no supporting work may receive no credit, while an incorrect answer with relevant work may receive partial credit. 1. (5 points extra credit) (a) This is my (b) I do semester at CU Boulder. percent of the textbook homework problems. 2. (25 points) Consider a solid cylinder of height H and radius R, in other words, the object bounded between the planes z = 0, z = H , and inside the cylinder x2 + y 2 = R2 . (a) If one were to cut through this cylinder with the plane x = a where 0 < a < R, determine the volume d of the smaller piece. (Hint: what is tan x?) dx (b) Does your result in part (a) make sense if a = 0? Explain. (c) Does your result in part (a) make sense if a = R? Explain. 3. (25 points) Use the transformation u = 3x + 2y and v = x + 4y to evaluate the integral I=
(3x2 + 14xy + 8y 2 ) dx dy
R over the region R in the ﬁrst quadrant bounded by the lines y = − 3 x + 1, y = − 3 x + 3, y = − 1 x, and 2 2 4 y = − 1 x + 1. 4 (a) Solve for x and y in terms of u and v using the given substitution. Be sure to check this because the rest of the problem depends on this result! (b) Transform the original region Rxy into its corresponding region Ruv in the uv -plane. Make a clear sketch of the new region of integration Ruv in the uv -plane. Be sure to label all axes, boundaries, intersection points, etc. on your sketch. (c) Rewrite the integral for I over the region Ruv in the uv -plane in terms of u and v . (d) Evaluate I in terms of u and v . 4. (25 points) The integral V= calculates the volume of an object.
2π √ 2 √ 4− z 2
θ =0 z =0 r =z r dr dz dθ (a) Make a clear sketch of the cross-section of the object in the rz -plane (this is a constant θ plane in cylindrical coordinates) clearly labeling the bounding surfaces of the region of integration. (If you have trouble with this, you may “buy” a sketch of the shape of the region in the rz -plane for 5 points. This sketch will only show the shape of the region, so you will still need to supply the remaining details. The oﬀer to buy this sketch ends at 7:00 pm!) (b) Express V in cylindrical coordinates using the order dz dr dθ. (c) Express V in spherical coordinates using the order dρ dφ dθ. √ √ if necessary, you may refer to (Hint: particular angles as φ = arctan(a/b). For example, φ = arctan( 2/ 3).) (d) Express V in spherical coordinates using the order dφ dρ dθ. (e) Evaluate one of the integrals above to determine the value of V . OVER 5. (25 points) Consider the path (section 1) starting on the x-axis at (1, 0), through the ﬁrst quadrant along the quarter-circle x2 + y 2 = 1 to the point (0, 1), then (section 2) straight down the y -axis from (0, 1) to the origin (0, 0), and the vector function given by F = (x − y ) i + (x + y ) j. (a) Sketch the entire path and give a parametrization for each section of the path. (b) Calculate the ﬂux along section 1 of the path. (c) Calculate the ﬂow along section 2 of the path. (d) If one, or both, of the calculations in parts (b) and (c) can be veriﬁed using another concept from Calculus III, then clearly state why and perform the necessary calculations. Otherwise state that neither result can be veriﬁed. Projections and distances projA B = Arc length, frenet formulas, and tangential and normal acceleration components ds = |v| dt dT = κN ds dB = −τ N ds T= dr v = ds |v | N= dT/ds dT/dt = |dT/ds| |dT/dt| A·B A A·A − → |P S × v | d= |v | − → P S · n d= |n| B=T×N τ =− dB ·N ds a N = κ| v | 2 = dt Directional derivative, discriminant, and Lagrange multipliers a = aN N + aT T aT = df = ( ∇f ) · u ds Polar coordinates x = r cos θ fxx fyy − (fxy )2 y = r sin θ r 2 = x2 + y 2 dT |f (x)| | xy − y x| ˙ ¨ ˙¨ = |v × a| = κ= =2 3 (x))2 |3/2 ds |v | | 1 + (f | x + y 2 | 3/ 2 ˙ ˙ d| v | |a|2 − a2 T ∇f = λ∇g, g=0 dA = dx dy = r dr dθ Cylindrical and spherical coordinates Cylindrical to Rectangular x = r cos θ y = r sin θ z=z Spherical to Cylindrical r = ρ sin φ z = ρ cos φ θ=θ Spherical to Rectangular x = ρ sin φ cos θ y = ρ sin φ sin θ z = ρ cos φ dV = dx dy dz = r dr dθ dz = ρ2 sin φ dρ dφ dθ Substitutions in multiple integrals f (x, y ) dx dy =
R G f (x(u, v ), y (u, v )) |J (u, v )| du dv Mass
C where δ dA
R J (u, v ) = ∂x ∂y ∂y ∂x − ∂u ∂v ∂u ∂v Mass, moments, and center of mass Moments Mx =
M= y δ dA
R My = x δ dA
R Center of mass
x = My /M ¯ y = Mx /M ¯ Flow and ﬂux Flow = C F · T ds = F · V dt =
C C F · dr =
C M dx + N dy
C Flux = F · n ds = M dy − N dx ...
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- Fall '07