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Unformatted text preview: MATH 1A: THE DEFINITE INTEGRAL 3 1. Approximate the definite integral than or greater than the actual value?
0 (2 - x)dx using six rectangles and right endpoints. Is this value less Can you find the exact value of the integral? (Hint: What does the graph look like?) 2 2. Approximate the integral
-2 4 - x2 dx (use your choice of the number of subintervals, left/ right endpoints or midpoints). What is the exact value of the integral? Date: November 20, 2009.
1 3. Use the definition of the definite integral (as a limit of the sum of areas of rectangles) to evaluate the following:
2 (x3 + 2x2 - x)dx.
0 The following identities may come in handy:
n 1 + 2 + + n =
i=1 n i= n(n + 1) . 2 n(n + 1)(2n + 1) . 6 n2 (n + 1)2 . 4 1 2 + 2 2 + + n2 =
i=1 n i2 = i3 =
i=1 1 3 + 2 3 + + n3 = 4. Find
- sin(x)dx. (Hint: Before you venture into a computation, see if you can deduce anything graph- ically.) 1 1 5. What's bigger:
0 x3 dx or 1? What about
0 x3 dx and 1 ? 3 2 ...
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This note was uploaded on 03/27/2012 for the course MATH 1a taught by Professor Benedictgross during the Fall '09 term at Harvard.
- Fall '09