Evaluate the integral 16 S² x² + 4 Your answer should be in the form kπ, where k is an integer. What is the value of k? Hint: k -dx. = darctan(x) dx = 1 x² + 1

Calculus: Early Transcendentals
8th Edition
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Author:James Stewart
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Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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(a)
Evaluate the integral
16
S²
x² + 4
Your answer should be in the form kл, where k is an integer. What is the value of k?
Hint:
k
=
ao
a1
(b)
Now, lets evaluate the same integral using power series. First, find the power series for the function
16
f(x)
a2
Then, integrate it from 0 to 2, and call it S. S should be an infinite series.
x² + 4
What are the first few terms of S?
a3
a4
||
||
||
-dx.
||
darctan(x)
dx
=
=
=
1
x² + 1
(c) The answers to part (a) and (b) are equal (why?). Hence, if you divide your infinite series from (b) by k
(the answer to (a)), you have found an estimate for the value of in terms of an infinite series.
Approximate the value of π by the first 5 terms.
Transcribed Image Text:(a) Evaluate the integral 16 S² x² + 4 Your answer should be in the form kл, where k is an integer. What is the value of k? Hint: k = ao a1 (b) Now, lets evaluate the same integral using power series. First, find the power series for the function 16 f(x) a2 Then, integrate it from 0 to 2, and call it S. S should be an infinite series. x² + 4 What are the first few terms of S? a3 a4 || || || -dx. || darctan(x) dx = = = 1 x² + 1 (c) The answers to part (a) and (b) are equal (why?). Hence, if you divide your infinite series from (b) by k (the answer to (a)), you have found an estimate for the value of in terms of an infinite series. Approximate the value of π by the first 5 terms.
(c) The answers to part (a) and (b) are equal (why?). Hence, if you divide your infinite series from (b) by k
(the answer to (a)), you have found an estimate for the value of in terms of an infinite series.
Approximate the value of π by the first 5 terms.
(d)
What is the upper bound for your error of your estimate if you use the first 6 terms? (Use the alternating
series estimation.)
Transcribed Image Text:(c) The answers to part (a) and (b) are equal (why?). Hence, if you divide your infinite series from (b) by k (the answer to (a)), you have found an estimate for the value of in terms of an infinite series. Approximate the value of π by the first 5 terms. (d) What is the upper bound for your error of your estimate if you use the first 6 terms? (Use the alternating series estimation.)
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