anonymous
  • anonymous
I can't believe sometimes that English is my first language. What does this question mean? "Here is a plot of f[x] = x^3 on [-2,2] How does the plot give away the value of Integrate[x^3, {x, -2, 2}] ?​" pictures coming ...
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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anonymous
  • anonymous
image attached
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UsukiDoll
  • UsukiDoll
ok... so we are given a function \[\LARGE f(x) = x^3\] graphed between the intervals of -2 to 2
anonymous
  • anonymous
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UsukiDoll
  • UsukiDoll
and then we are asked how the plot of that graph gave away the value when we are integrating from -2 to 2
UsukiDoll
  • UsukiDoll
you wouldn't mind if we integrate first right? do you know anti-derivatives?
ganeshie8
  • ganeshie8
Hint : if \(f(x)\) is an odd function, \[\int\limits_{-a}^a f(x)\,dx = 0\]
anonymous
  • anonymous
I dont mind, but negative on the anti derivatives (I think)
UsukiDoll
  • UsukiDoll
when we are taking the antiderivative, we add one to the exponent and divide by the new exponent.
UsukiDoll
  • UsukiDoll
so since your exponent is 3. What's 3 +1 ?
anonymous
  • anonymous
so do they mean, I am to look at the plot, and then work out the value of the integration equation?
anonymous
  • anonymous
hmmm.. 5-1?
anonymous
  • anonymous
j/k 4
UsukiDoll
  • UsukiDoll
|dw:1436955882543:dw| now evaluate f(b)-f(a) or f(2)-f(-2)
anonymous
  • anonymous
when the question says.. the value of ... do they mean that the equation itself is a value, or that the equation comes out to some value?
ganeshie8
  • ganeshie8
Notice that the graph is symmetric about origin, so f(x) is an odd function. You don't need to do any further work, the definite integral between a symmetric interval would be simply 0 because the positive area and negative area kill each other out.
anonymous
  • anonymous
oh, so by looking at the image.. I should see that it is symmetrical about zero, and therefore the sum of it's parts will be a negative area + an equal positive area = zero.
UsukiDoll
  • UsukiDoll
yes it is an odd function, so we do have symmetry around the origin . When we graph x^3 only we should have just |dw:1436956009479:dw|
UsukiDoll
  • UsukiDoll
but keep in mind we are restricted from -2 to 2 as well
ganeshie8
  • ganeshie8
Exactly!
anonymous
  • anonymous
thank you both.. gotcha, I just wasn't sure why they would ask me about.. 'giving away' .. I was like.. "giving away what?"
anonymous
  • anonymous
The author of this program uses so many strange metaphors to me
UsukiDoll
  • UsukiDoll
|dw:1436956101513:dw| well the result does come to 0 if we do evaluate after taking the anti-derivative.
UsukiDoll
  • UsukiDoll
so there is cancellation...
ganeshie8
  • ganeshie8
Indeed, the graph does give away the fact the the given function is "odd" |dw:1436956075476:dw|
UsukiDoll
  • UsukiDoll
so by having an odd function, there's symmetry around the origin, the negative and positive cancel out and that leads to a 0 which is the same answer if we were able to evaluate that integral. By taking the anti-derivative and evaluating... we will have 0 (though I like to work backwards....easier)
UsukiDoll
  • UsukiDoll
maybe the cancellations gave out the answer to that integral ?
ganeshie8
  • ganeshie8
You could also work it analytically : \(f(-x) = (-x)^3 = -x^3 = -f(x)\) \(f(-x) = -f(x)\) means \(f(x)\) is an odd function.
anonymous
  • anonymous
gotcha, I haven't got to anti derivatives yet.. I think they're coming up.. and they just want me to look at the graph.. there's another identical one next..
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UsukiDoll
  • UsukiDoll
another odd function... thanks to symmetry on the origin
ganeshie8
  • ganeshie8
same thing, just check if it is an odd one
anonymous
  • anonymous
hmmm, they might want me to do it analytically too..
UsukiDoll
  • UsukiDoll
I don't think we need integration by parts on this one... just u-substitution.
ganeshie8
  • ganeshie8
|dw:1436956504186:dw|
anonymous
  • anonymous
Well there is that f[-x] == -f[x] symmetry going on.
anonymous
  • anonymous
yeah, you must be telepathic
UsukiDoll
  • UsukiDoll
xD!
anonymous
  • anonymous
thank you for your instant help guys.. I dont think it was 30 seconds and you had me covered.
anonymous
  • anonymous
please someone approve usukiDoll for qualified helper status.
UsukiDoll
  • UsukiDoll
I was going to do my nails when I saw the QH Questions change color, so I was like someone just asked a question.
UsukiDoll
  • UsukiDoll
Good thing I didn't have wet polish otherwise I would have to do it over XD
anonymous
  • anonymous
Do all odd functions center on origin? or is this only because x^3 is fairly basic. And by odd did you mean the expression had degree 3, or odd means any function of an odd degree?
UsukiDoll
  • UsukiDoll
if the function is odd then there is symmetry at the origin and yes we need an odd exponent number like \[\large f(x) = x^3,f(x)=x^5.f(x)=x^7\]
UsukiDoll
  • UsukiDoll
there's also a test too.. that determines if the function is odd or not
UsukiDoll
  • UsukiDoll
now if we have an even function, that means that there is symmetry on the y-axis the parabola \[\large f(x) =x^2\] is the most common example for an even function
UsukiDoll
  • UsukiDoll
for trigonometry , sinx cosx, tanx sinx and tanx are odd functions cosx is the only even function
anonymous
  • anonymous
So then it is only functions then that exhibit the property of -f[x] + f[x] == 0 that are considered odd?
ganeshie8
  • ganeshie8
For simple polynomials we do need odd degree, but that is not so important. It would be more simple to just stick to the definition : \(f(x)\) is an odd function if \(f(-x) = -f(x)\)
UsukiDoll
  • UsukiDoll
erm.. you have to plug in -x in your x and yes the definitions of odd and even functions help a lot.
UsukiDoll
  • UsukiDoll
So, take the function and plug in -x for x and simplify should our result become \[\large f(-x)= f(x) \] the function is even and are signs are the same If our signs are the opposite (for example minus signs become plus signs or plus signs become minus signs) then we have an odd function \[\large f(-x)=-f(x) \] There is a chance that our function is neither odd nor even
ganeshie8
  • ganeshie8
for example below graph is odd too; there is no algebraic expression to express it though |dw:1436958097638:dw|