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20b_03f_2e - (a dy dx = e x y y(0 = 0(b y ′ t-t y t 2 =...

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Math 20B Second Midterm 50 points November 20, 2003 Print Name, ID number and Section on your blue book. BOOKS and CALCULATORS are NOT allowed. One sheet of NOTES is allowed. You must show your work to receive credit. 1. (8 points) Which of the following integrals diverge? Remember to give a reason for your answer in each case! (a) integraldisplay 1 0 e sin x dx (b) integraldisplay 1 0 dx x 2. (12 points) Consider the curve given by y ( x ) = ln x for 1 x e . (a) Write down an integral for its length. (b) The curve is rotated about the y -axis. Write down an integral for the surface area. Do NOT evaluate the integrals. 3. (16 points) Solve the differential equations:
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Unformatted text preview: (a) dy dx = e x + y , y (0) = 0. (b) y ′ ( t )-t ( y ( t )) 2 = t (general solution). 4. (8 points) Write down an integral in polar coordinates for the area of the region that lies inside the curve r = 2cos θ and outside the curve r = √ 2. 5. (6 points) Consider the diFerential equation y ′ ( t ) = 1-y 2 . ±ind the limiting behavior of y ( t ) (that is, what is lim t → + ∞ y ( t )) if the initial condition is (a) y (0) = 0 (b) y (0) = 2. You do not need to solve the equation — a clear explanation in a few words will su²ce. END O± EXAM...
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