Please answer all questions below in their entirety in the easiest way to unders
ID: 2870436 • Letter: P
Question
Please answer all questions below in their entirety in the easiest way to understand. Each question has multiple parts and also in question 4 you are to choose multiple choice answers, so there should be 2 answers to each part of question 4. Thanks in advance!
Question 1
Evaluate -44 (16-x2)1/2 dx
(a) Trapezoidal, n = 8: TS = ?
(b) Simpson's, n = 8: SS = ?
(c) Exact: = ?
Question 2
Suppose you estiamate 03 (x+1)-1/2 dx using numerical methods.
Estimate the error with the number of subintervals n = 10 for
a) Trapezoidal Rule: |ET| ?
b) Simpson's Rule: |ST| ?
Question 3
Suppose you estiamate 13 xlnx dx using numerical methods.
Find the number of subintervals n if the error is to be within 0.001.
a) Trapezoidal Rule: n ? (Provide an integer)
b) Simpson's Rule: n ? (Provide an integer)
Question 4
1) -11 1/(1+x2) dx is
a.) not an improper integral or
b.) an improper integral
and
a.) converges
b.) not applicable (if not an improper integral)
c.) diverges
2) -11 1/(x2) dx is
a.) not an improper integral or
b.) an improper integral
and
a.) not applicable (if not an improper integral)
b.) diverges
c.) converges
3) --10 1/x dx is
a.) not an improper integral or
b.) improper integral
And
a.) diverges
b.) converges
c.) not applicable (if not an improper integral)
4) 2 (x-2)-3/2 dx is
a.) not an improper integral or
b.) an improper integral
and
a.) diverges
b.) converges
c.) not applicable (if not an improper integral)
Question 5
a) 0 e-x(cos x) dx = ?
b) -1 (2-x)-3/2 dx = ?
c) - 1/(9x2+1) dx= ?
Explanation / Answer
a = lower limit = -4
b = upper limit = 4
n = 8
h = (b - a)/n = 1
So, endpoints are -4 , -3 , -2 , -1 , 0 , 1 , 2 , 3 and 4
f(-4) = (16 - (-4)^2)^(1/2) = 0
f(-3) = (16 - 9)^1/2 = sqrt(7)
f(-2) = sqrt(12)
f(-1) = sqrt(15)
f(0) = 4
f(1) = sqrt(15)
f(2) = sqrt(12)
f(3) = sqrt(7)
f(4) = 0
a) trapezoidal :
Now, for the trap rule, add em like this :
f(-4) + 2(f(-3) + f(-2) + f(-1) + f(0) + f(1) + f(2) + f(3)) + f(4)
So, it becomes :
0 + 2(sqrt(7) + sqrt(12) + sqrt(15) + 4 + sqrt(15) + sqrt(12) + sqrt(7)) + 0
47.9313450896390483
And now, we multiply by h/2 :
47.9313450896390483 * 1/2
23.96567254481952415 ----> FIRST ANSWER
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b) Simpson'r rule :
We add like this :
f(-4) + 4f(-3) + 2f(-2) + 4f(-1) + 2f(0) + 4f(1) + 2f(2) + 4f(3) + f(4)
0 + 4sqrt(7) + 2sqrt(12) + 4sqrt(15) + 2(4) + 4sqrt(15) + 2sqrt(12) + 4sqrt(7) + 0
74.0062837187270782
And multiply by h/3 :
74.0062837187270782 * 1/3
24.6687612395756927333 ---->SECOND ANSWER
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c) Exact answer :
When we graph, we see it that it is a semicircle with radius = 4
So, area = pi * r^2 / 2
A = pi * (4)^2 / 2
A = 8pi ----> EXACT ANSWER