M 408D-Ariel-Test 2

# M 408D-Ariel-Test 2 - 0.86 1/2'if/2 1 3 Consider the...

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M408D - Calculus II: Midterm 2 Instructor: Gil Ariel Write your name, EID and the unique number you are registered to. Answer all questions and show work. The posted diff.Zint. table + 1 formula sheets are allowed. Calculators are not allowed. Good luck. 1. Find the area of the region that is bounded by the curve r =..; sin e and lies in the sector 0 :s: e :s: 'if. 2. Consider the following curve given by the polar equation r = 3 sin e, 0 :s: e :s: 'if. (a) Sketch the curve. You can use the table below. (b) Show that the curve satisfies the equation of a circle in Cartesian coordinates. What is its center and radius? e sine cose 0 0 1 'if/6 1/2 V3/2 = 0.86 'if/4 1/..;2 1/..;2 'if/3 V3/2 = 0.86 1/2 'if /2 1 0 3. Consider the following curve:
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Unformatted text preview: 0.86 1/2 'if/2 1 3. Consider the following curve: r(t) = t 2 i + sin t3 + costk. (a) Find the tangent vector to the curve at time t, T(t) . (b) Write a vector equation for the plane that is perpendicular to the curve at t = 'if/4. (c) Find the length of the part of the curve that lies inside the sphere x2 + y2 + z2 = 17. Express your answer in the form of an integral. Hint: start by finding the points of intersection ,between the curve and the sphere. 4. Find the length of the loop of the curve x = 3t -t 3 , Y = 3t2. 1...
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