Spring 2004 - Nordgren's Class - Quiz 3

# Spring 2004 - Nordgren's Class - Quiz 3 - Solutions to quiz...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

Spring 2004 - Nordgren's Class - Quiz 3 - Solutions to quiz...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online