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Unformatted text preview: Solutions to quiz 3
By H akan Nordgren Question 1: Evaluate the indefinite integral sin3 x cos2 xdx. Answer: We are going use the fact that sin3 x = sin x(1 - cos2 x). Let u = cos x. Then du = - sin xdx. Thus sin3 x cos2 xdx = = - = - sin x(1 - cos2 x) cos2 xdx (1 - u2 )u2 du (u2 - u4 )du 1 1 = - u3 + u5 + constant 3 5 1 1 = - cos3 x + cos5 x + constant 3 5 Question 2: Explain with the aid of formulae and a sketch the midpoint rule for b estimating a f (x)dx using n subintervals of length x = b-a and points x0 , . . . , xn . n Answer: Again, the diagrams you will have to supply yourselves. A formula that should be mentioned is
b a f (x)dx x [f (x1 ) + . . . + f (xn )] , where xi = 1 (xi - xi-1 ). 2 Question 3: Determine whether the integral converges find its value.
4 dx 0 x is convergent or divergent; if it 1 Answer: The first thing to do is to determine where x is badly behaved on the interval [0, 4]. The problem is at the zero end of the interval, because the function 1 will have us dividing by 0. So to evaluate this integral we write x 4 0 dx = x dx t0 x t 4 = lim 2 x - lim -
4 t0 t = 2(4 - 0) = 8. Thus the integral converges to the value 8. 1 Question 4: Let y1 (x) = x and let y2 (x) = x2 . Find the values of x for which y1 and y2 intersect, sketch the region between the curves, and find the area of the region between the curves. Answer: The curves will intersect when y1 (x) = y2 (x); that is, when x = x2 . This happens when x(x - 1) = 0, or when x = 0, 1. The area of the region between the curves is
1 0 (x - x2 )dx = 1 2 1 3 x - x 2 3 1 1 - = 2 3 1 = 6 1 0 2 ...
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- Spring '08
- Calculus, Berlin U-Bahn, integral converges, akan Nordgren Question, indefinite integral sin3