Question

So I have a graph of f(t) and then I have the function F(x)=∫f(t)dt. I do not know how to put in the bounds here to the function F(x)=the integral from 3 to x of f(t). The graph is below

https://ka-perseus-graphie.s3.amazonaws.com/cd8014d99190a17357649439d26c805f9da21e34.png

I need to know the values of F(0), F(-6), F(-2), F(8) and F(3). I know F(3)=0, but I can't seem to figure out the other ones. Any help is much appreciated!

Answer 1

Well \(f(x)\) is a piece-wise function. You can define it in 3 parts:
1) for \(-6 \le x \le -2\)
2) for \(-2 < x \le 3\)
3) for \(3 < x \le 8\)

Write what \(f(x)\) is for each of these intervals. Then, when you want to find the integral \(\large F(x)=\int\limits_{-\infty}^x f(t) \, dt\) you will have to break it into 2 integrals if you are trying to find F(x) when x is between -2 and 3 , and into 3 integrals if you are finding F(x) if x is between 3 and 8

Answer 2

Just as an example: \[\large F(0)=\int_{-\infty}^0f(x)dx=\int_{-6}^{-2}f(x)dx+\int_{-2}^0f(x)dx \]

Answer 3

\[\large f(x) = \begin{cases} 4&& -6\le x < -2 \\-2x&& -2 \le x < 3\\-6&& 3 \le x \le 8\end{cases} \]

Answer 4

Also, important to know, is that the integral of a single point (or thinking of it as the integral with the lower and upper bounds being the same) is 0:
If you have a function which is only defined on some interval \([a, b], b>a\) and is 0 elsewhere, then:
\[\large F(a)=\int_{a}^{a}f(x)dx=0 \]