548
Chapter 5 | Integration
142. If f is 1-periodic ⎛ integrable over [0, 1],
⎝ f ( t +1) = f ( t ) ⎞
⎠ , odd, and
137. [T] Compute the left and right Riemann sums, L 10 and R 10 , and their average L 10 + R 10 2 for f ( t ) = ⎛ ⎝ 4− t 2 ⎞ ⎠
is it always true that
1
∫
over [1, 2]. Given that ∫ 1 2 ⎛
⎞ ⎠ dt =1.66 –
f ( t ) dt =0?
⎝ 4− t 2
, to how
0
L 10 + R 10 2
143. If f is 1-periodic and ∫ 0 1
many decimal places is
accurate?
f ( t ) dt = A , is it
1+ a
5
necessarily true that ∫ a
∫
1+ t 4 dt = 41.7133...,
f ( t ) dt = A for all A ?
138.
If
what
is
1
5
∫
1+ u 4 du ?
1
139. Estimate ∫ 0 1 tdt using the left and right endpoint sums, each with a single rectangle. How does the average of these left and right endpoint sums compare with the actual value ∫ 0 1 tdt ? 140. Estimate ∫ 0 1 tdt by comparison with the area of a single rectangle with height equal to the value of t at the midpoint t = 1 2 . How does this midpoint estimate compare with the actual value ∫ 0 1 tdt ? 141. From the graph of sin(2 πx ) shown: a. Explain why ∫ 0 1 sin(2 πt ) dt =0. b. Explain why, in general, ∫ a a +1 sin(2 πt ) dt =0 for any value of a .
This OpenStax book is available for free at http://cnx.org/content/col11964/1.12
Made with FlippingBook - professional solution for displaying marketing and sales documents online