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I was doing my homework last night, and 1 problem decided that it would not let me solve it. =/
f(x) = |x2 -4x + 3| Integrate from [0, 4]
(Don't know how to insert the integral symbol, hope you understand it the way I wrote the problem.
It's the last question in the set, and I just can't get it for some reason.
What's tripping you up on this problem? It is pretty straightforward since there is no compositions or other complexities in the polynomial. Simply integrate, plug in 4 and 0, and find the difference.
The final answer is 4/3, so try to work toward that. What specific step is causing you problems?
edit:
Whoops, I totally missed the absolute value.
The standard procedure for integrating absolute value functions is to find whatever ranges within the given range where the function is negative, and simply negate those ranges when you sum them all up at the end.
In this function, there exists a negative range from [1,3], so you'll have to break it up into three separate integrals, from ranges [0,1], [1,3], and [3,4], and remember to negate the result of your [1,3] range before adding the three results up.
Let me know if you're still having trouble.
Absolute sign means sqrt(f(x)^2)
So it'll become:
F(x) = Int[ sqrt((x^2 - 4x + 3) (x^2 -4x + 3)) dx] 0-4
F(x) = Int[ sqrt( x^4 - 8x^3 + 22x^2 - 24x + 9) dx] 0-4
F(x) = (x^4 - 8x^3 + 22x^2 - 24x + 9)^1.5 * 1/1.5 * 1/(4x^3 - 24x^2 + 44x - 24) |0-4
F(0) = 9^1.5 * 1/1,5 * -1/24 = 9 * 3 * 2/3 * -1/24 = -18/24 = -3/4
F(4) = (0 + 9)^1.5 * 1/1.5 * 1/(24) = 3/4
F(4) - F(0) = 3/4 - (-3/4) = 1.5
...Necro's way might be easier, and less prone to mistakes
I found my mistake. When finding the zero's of the parabola, I wrote a 3 instead of a 2, so my values where I broke up the integral were wrong. -_-
Also AK you made a mistake somewhere. Answer is 4.
As I said:Quote:
Originally Posted by qwertyxsora
xDQuote:
Originally Posted by Aerith's Knight
Also, don't you mean 2 instead of 3, as the zero's of the parabola are 1 and 3.
I just checked in Mathematica, and the
F(x) = Int[ sqrt( x^4 - 8x^3 + 22x^2 - 24x + 9) dx] 0-4
part is still good, but when I looked at the answer for the integral that filled up the entire page... yeah, you really should use Necro's way. xD
If you split them up, the parts just become:
Int [x^2 - 4x + 3 dx]0-1 = 1 1/3
Int [-x^2 + 4x - 3 dx]1-3 = 1 1/3
Int [x^2 -4x + 3 dx]3-4 = 1 1/3
--------------------------------- +
Total = 4
It's good to know both ways though, if you can't find the zero points that easily, like with for example: "x^2 -x -1"