Please answer all questions below in their entirety in the easiest way to unders

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

--------------------------------------------------------------------------------------------

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

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