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Math Help - confusion with a question about integrals, multiple choices

  1. #1
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    Question confusion with a question about integrals, multiple choices+questions

    Hi.

    Given

     \int_{-1} ^2 {f(x) =a} and  \int_0 ^2 {f(x) = b

    the value of
     \int_0 ^{-1} f(x) is:

    A. a+b
    B. 2a-b
    C. a-b (which I think is right)
    D. a-2b
    E. b-a (might be true?)

    I am confused because the last integral boundaries are from 0 to -1 and I always see them written from the smaller to the bigger number, aka from -1 to 0.
    Does this makes a the result different? (maybe it's a mistake in the printing?)

    Thanks.


    attached image is another question, sorry for the unprofessional editing, it's from the book and it's not in English so i had to edit it a little... the pen markings are my own, what I think are correct/incorrect.
    So I basically think statements 2 and 3 are correct, therefore answer D would be my choice, what do you think?

    I need to send this out today and have total of 18 problems to answer (luckily this time they are multiple choices, not "open ended")
    I would probably add more pictures as it's much easier and faster than typing the whole equations in the forum...

    Thanks.
    Attached Thumbnails Attached Thumbnails confusion with a question about integrals, multiple choices-untitled.png  
    Last edited by ryu1; May 26th 2013 at 03:32 PM.
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  2. #2
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    Re: confusion with a question about integrals, multiple choices+questions

    Quote Originally Posted by ryu1 View Post
    Given
     \int_{-1} ^2 {f(x) =a} and  \int_0 ^2 {f(x) = b
    the value of
     \int_0 ^{-1} f(x) is:

    \int_{ - 1}^0 {f}  + \int_0^2 {f}  = \int_{ - 1}^2 {f} and \int_{ - 1}^0 {f}  =  - \int_0^{ - 1} {f}

    SO?
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  3. #3
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    Re: confusion with a question about integrals, multiple choices+questions

    Quote Originally Posted by Plato View Post
    \int_{ - 1}^0 {f}  + \int_0^2 {f}  = \int_{ - 1}^2 {f} and \int_{ - 1}^0 {f}  =  - \int_0^{ - 1} {f}

    SO?
    so it's E. b-a

    \int_{ - 1}^0 {f} [= z]  + \int_0^2 {f} [= b]   = \int_{ - 1}^2 {f} [= a]

    z+b=a
    z=a-b
    -z=b-a

    right?
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  4. #4
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    Re: confusion with a question about integrals, multiple choices+questions

    Quote Originally Posted by ryu1 View Post
    so it's E. b-a

    \int_{ - 1}^0 {f} (= z)  + \int_0^2 {f} (= b)   = \int_{ - 1}^2 {f} (= a)

    z+b=a
    z=a-b
    -z=b-a

    right?
    Correct!
    Thanks from ryu1
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  5. #5
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    Re: confusion with a question about integrals, multiple choices+questions

    Quote Originally Posted by Plato View Post
    Correct!
    Thank you very much!

    What you think about the problem in the picture?

    Thanks.
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  6. #6
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    Re: confusion with a question about integrals, multiple choices+questions

    Hello, ryu1!


    Attached image is another question.
    So I basically think statements 2 and 3 are correct.
    Therefore, answer D would be my choice. .What do you think? . I agree!

    Code:
          |
          *
          |   *
          |   |::*
          |   |::::*      4 
        --+---+------*----+----
          |   1      3*:::|
          |             *:|
          |               *
    The graph of f(x) is continuous and passes through (3,0).
    S = shaded area.
    F(x) is the antiderivative of f(x).

    1.\;S \:=\:\int^4_1 f(x)\,dx \qquad\qquad\qquad\qquad 2.\;\int^4_1f(x)\,dx \:=\:F(4) - F(1)

    3.\;S \:=\:\int^3_1f(x)\,dx - \int^4_3f(x)\,dx \qquad 4.\;S \:=\:F(4) - F(1)


    Which are the correct statements?

    \begin{array}{cccccccccccc}(A)& 2 && (B) & 1,3 && (C) & 2,4 && (D) & 2,3 \\ \\ (E)&1,2,4 && (F)&1 && (G) &1,2,3 && (H) & 3 \end{array}

    Statement 1 is incorrect.
    The integral from 1 to 4 is not the shaded area.
    It is the net area (some of which is negative).

    Statement 4 is equivalent to statement 1.
    It is also incorrect.

    Statement 2 is correct.
    It gives the value of the definite integral.
    \int^4_1f(x)\,dx \:=\:F(x)\bigg]^4_1 \:=\:F(4)-F(1)

    Statement 3 is correct.
    The total shaded area is the integral from 1 to 3
    . . minus the integral from 3 to 4.
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  7. #7
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    Re: confusion with a question about integrals, multiple choices+questions

    Quote Originally Posted by Soroban View Post
    Hello, ryu1!



    Statement 1 is incorrect.
    The integral from 1 to 4 is not the shaded area.
    It is the net area (some of which is negative).

    Statement 4 is equivalent to statement 1.
    It is also incorrect.

    Statement 2 is correct.
    It gives the value of the definite integral.
    \int^4_1f(x)\,dx \:=\:F(x)\bigg]^4_1 \:=\:F(4)-F(1)

    Statement 3 is correct.
    The total shaded area is the integral from 1 to 3
    . . minus the integral from 3 to 4.
    Thank you.
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  8. #8
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    Another problem

    New problem:
    Given the function confusion with a question about integrals, multiple choices-func.png, R is a constant.

    Find the anti derivative F(x) that such that F(R) = 0.

    What is the value of F(0.5R)+F(2R)+F'(0.5R)+F'(2R)?

    Answers:
    confusion with a question about integrals, multiple choices-ans.png

    I tend to go with C because I somehow got to 3/2R after calculating only the F'(0.5R)+F'(2R) (these are just the given function right?)
    But I have a feeling the F(0.5R)+F(2R) will change it to something else...also where does the 0.19 comes from I wonder.

    Thanks
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