# Parenrightbigg 2 parenleftbigg y 1 2 parenrightbigg 2

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parenrightbigg 2 + parenleftbigg y - 1 2 parenrightbigg 2 = 1. (b) Evaluate integraldisplay γ F · d s , where F ( x, y ) = ( x + 2 y, x 2 - y 3 ) and γ is a parametrization of the line segments from (1 , 1) to (3 , 1) to (3 , - 1). (c) Evaluate integraldisplay γ ω , where ω = 2( x + y 2 ) dx + 4 xy dy and γ is a parametrization of the polar graph r = 2 sin parenleftbigg θ 6 parenrightbigg from θ = 0 to θ = π .

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MATB42H page 2 5. [5 points] Use Green’s Theorem to give a line integral which is equivalent to the double integral integraldisplay 3 - 3 integraldisplay 2 q 1 - x 2 9 - 2 q 1 - x 2 9 3 x 2 y 2 dy dx . (You are not required to evaluate the inte- gral.) 6. [6 points] Use a path integral to compute the area of the first quadrant region bounded by a piece of the astroid x 2 / 3 + y 2 / 3 = 1 and the axes. 1 x 1 y 7. [5 points] Evaluate the surface integral integraldisplay S ( x + z ) dS , where S is the first octant portion of the cylinder y 2 + z 2 = 9 between x = 0 and x = 4. 0 1 2 3 4 x 0 1 2 3 y 0 1 2 3 z 8. [32 points] (a) Let ω = z r 3 dx dy + x r 3 dy dz + y r 3 dz dx where r = radicalbig x 2 + y 2 + z 2 and let R be the solid ellipsoid x 2 4 + y 2 9 + z 2 16 1. Evaluate integraldisplay ∂R ω . (b) Evaluate integraldisplay S (curl F ) · d S , where F ( x, y, z ) = ( e z 2 , 4 z - y, 8 x sin y ) and S is the upper hemisphere of radius 2 ( x 2 + y 2 + z 2 = 4, z 0) oriented by the unit normal which points away from the origin.
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