PracExam2 - in the counterclockwise direction from the...

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Math 32b Practice second hour exam. 1.Let B be the region described by x 0 , y 0 , z 0 , 1 2 x 2 + y 2 + z 2 1 . a) Sketch this region, and describe it in terms of spherical co-ordinates ρ, θ, φ. b) Use spherical co-ordinates to evaluate Z Z Z B q x 2 + y 2 + z 2 dV. Hint: ± ± ± ( x,y,z ) ( ρ,θ,φ ) ± ± ± = ρ sin 2 φ. 2. Let ( x, y ) = T ( u, v ) be define by x = u 2 - v 2 , y = 2 uv . a) If R is the region described by 1 u 2 + v 2 4 , u 0 , v 0 , sketch the region E = T ( R ) . b) Use this change of variable to evaluate the integral Z Z D dxdy x 2 + y 2 . Include a calculation of the Jacobian determinant. 3. a) Sketch the vector field ~ F ( x, y ) = y ~ i - x ~ j . b) Determine R ~ F · d~ r where C is the unit circle at the origin traversed
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Unformatted text preview: in the counterclockwise direction from the positive X axis to the negative X axis. 4. Determine the following line integrals: a) R ydx + x 2 dy, where is given by x = t, y = t 2 ,-1 t 2 . Include a sketch of the curve. b) R T ydx + x 2 dy where T is the triangle determined by the points (0 , 0), (1 , 2) , (-1 , 3) traversed in the clockwise direction (include a sketch of the curve with arrows indicating the direction of the motion). 1...
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