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Draw a diagram to interpret this equation geometrially as an equality

  1. Apr 23, 2005 #1
    I need some help with the following problems. Any help is highly appreciated.

    1. If [tex]f[/tex] is continuous on [tex]\mathbb{R}[/tex], prove that

    [tex]\int _a ^b f(-x) \: dx = \int _{-b} ^{-a} f(x) \: dx[/tex]

    For the case where [tex]f(x) \geq 0[/tex] and [tex]0 < a < b[/tex], draw a diagram to interpret this equation geometrially as an equality of areas.

    2. If [tex]f[/tex] is continuous on [tex]\mathbb{R}[/tex], prove that

    [tex]\int _a ^b f(x + c) \: dx = \int _{a+c} ^{b+c} f(x) \: dx[/tex]

    For the case where [tex]f(x) \geq 0[/tex], draw a diagram to interpret this equation geometrially as an equality of areas.

    3. If [tex]a[/tex] and [tex]b[/tex] are positive numbers, show that

    [tex]\int _0 ^1 x^a (1 - x) ^b \: dx = \int _0 ^1 x^b (1 - x) ^a \: dx[/tex]

    Here is what I've got so far:

    1. Consider the left-hand side

    [tex]\int _a ^b f(-x) \: dx[/tex]

    and apply the substitution rule:

    [tex]u=-x \Rightarrow \frac{du}{dx} = -1 \Rightarrow dx = - du[/tex]

    [tex]u(b)=-b[/tex]

    [tex]u(a)=-a[/tex]

    [tex]\int _a ^b f(-x) \: dx = -\int _{-a} ^{-b} f(u) \: du = \int _{-b} ^{-a} f(u) \: du = \int _{-b} ^{-a} f(x) \: dx[/tex]

    2. Consider the left-hand side

    [tex]\int _a ^b f(x + c) \: dx[/tex]

    and apply the substitution rule:

    [tex]u=x+c \Rightarrow \frac{du}{dx} = 1 \Rightarrow dx = du[/tex]

    [tex]u(b)=b+c[/tex]

    [tex]u(a)=a+c[/tex]

    [tex]\int _a ^b f(x + c) \: dx = \int _{a+c} ^{b+c} f(u) \: du = \int _{a+c} ^{b+c} f(x) \: dx[/tex]

    3. Consider the left-hand side

    [tex]\int _0 ^1 x^a (1 - x) ^b \: dx[/tex]

    and apply the substitution rule:

    [tex]u=1-x \Rightarrow \frac{du}{dx} = -1 \Rightarrow dx = -du[/tex]

    [tex]u(1)=0[/tex]

    [tex]u(0)=1[/tex]

    [tex]\int _0 ^1 x^a (1 - x) ^b \: dx = \int _1 ^0 u^b (1 - u) ^a \: du = \int _0 ^1 x^b (1 - x) ^a \: dx[/tex]
     
    Last edited: Apr 23, 2005
  2. jcsd
  3. Apr 23, 2005 #2

    arildno

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    This looks fine to me.
    Where do you need help?
     
  4. Apr 23, 2005 #3
    Oh, good! Thanks for checking it out.

    Other than that, I need to take care of the diagrams (problems 1 & 2). I'm not so sure how to handle those. It seems that the areas in #1 are symmetric with respect to the y-axis.
     
    Last edited: Apr 23, 2005
  5. Apr 23, 2005 #4

    arildno

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    You're right about diagram 1.
    For diagram 2, remember that f(x+c) represents a translation.
     
  6. Apr 23, 2005 #5
    I've got it. Thanks!
     
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