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Unformatted text preview: Math 20B Final Exam Fall 2003 • Print Name, ID number and Section on your blue book. • BOOKS and CALCULATORS are NOT allowed. Two sheets of NOTES are allowed. • You must show your work to receive credit. 1. (36 pts.) For each of the following integrals, either evaluate the integral or prove that it diverges. (a) integraldisplay e 1 x 3 ln x dx (b) integraldisplay π/ 2 2 + cos x x dx (c) integraldisplay (1 + 2 u ) 10 du (d) integraldisplay 1 / √ 2 x 2 √ 1- x 2 dx (e) integraldisplay 2 1 1 u + u 2 du (f) integraldisplay e t + e t dt 2. (9 pts.) Express the following in the Cartesian form a + bi or the polar form ( r, θ ), as indicated. Do not leave trig functions in your answers. (a) The values of w = 1 + i 1- 3 i and w in Cartesian form. (b) The values of (1- i ) 1 / 3 in polar form. (c) The value of e 2+ πi/ 4 in Cartesian form. 3. (15 pts.) The equation x 4 + y 4 = 1 describes a curve that looks somewhat like a square with rounded corners. Set up integrals for each of the following.square with rounded corners....
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