48 samplemid2

# 48 samplemid2 - coordinates 6 Section 7.1 6,12 Section 7.2...

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Practice Mid-term Two Math 20E Section B, Winter 2011 UCSD Name: ID Number: Section : Instructions: NO CALCULATOR. CLOSED BOOK, CLOSED NOTES, except for the formula sheet attached ID WILL BE CHECKED. GET IT READY! SHOW ALL YOUR WORK! (The ﬁnal answer without steps only earns partial credits.) 1. Let C ( t ) = ( t,t sin t,t cos t ) with 0 t π . Find the arc length. 2. Show that F = y cos x i + x sin y j is NOT a gradient vector ﬁeld 3. Find the integral of Z 1 0 Z 1 x Z 1 y ye z 4 dzdydx. Hint: change order of integral 4. Find volume of the region inside the ball centered at origin with radius a and the elliptic cylinder 2 x 2 + z 2 = 1 . 5. Find volume of region inside unit sphere and above plane z = 0 using triple integral in spherical

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Unformatted text preview: coordinates. 6. Section 7.1: 6,12 Section 7.2: 17 1 Formula Sheet (please do not remove this page from the test packet) • Arc length of c ( t ) L = Z b a k c ( t ) k dt • div curl F = and ∇ × ( ∇ f ) = 0. • Cavalieri’s Principle V = Z b a A ( x ) dx, where A ( x ) is the cross section area. • Change of variables in polar coordinates Z Z D f ( x,y ) dxdy = Z Z D * f ( r cos θ,r sin θ ) r dr dθ. • Change of variables in spherical coordinates Z Z Z W f ( x,y,z ) dxdy dz = Z Z Z W * f ( ρ sin φ cos θ,ρ sin φ sin θ,ρ cos φ ) ρ 2 sin φdρdθ dφ. 2...
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48 samplemid2 - coordinates 6 Section 7.1 6,12 Section 7.2...

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