find int[0,4] f(x) dx, for "f(x)= n" if n ≤ x < n + 1
- Thread starter lander
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In the "header you wrote:
f(x)= n if n =< x > n + 1
But in your text, you wrote:
f(x) = n if n ≤ x < n + 1 with n ∈ Z.
Which one is correct?
What are your thoughts?
Please share your work with us ...even if you know it is wrong.
If you are stuck at the beginning tell us and we'll start with the definitions.
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f(x) = n if n ≤ x < n + 1 with n ∈ Z. This one is correct.
Actually i do not really have an idea how to start to solve the problem.
Well, let's try graphing the function and see what we get. The problem statement defines f(x) as:
\(\displaystyle f(x) = n \text{ if } n \le x \le n + 1\)
Considering now other information is given, I'll assume that the function is undefined everywhere else. Now, let n = 0. \(\displaystyle f(x) = 0 \text{ if } 0 \le x \le 1\). What does the graph of that look like? Now let n = 1. \(\displaystyle f(x) = 1 \text{ if } 1 \le x \le 2\). What does that graph look like? Continue drawing n = 2, n = 3, and n = 4. Now go back and fill in some of the in between points, like say n = 0.25, n = 0.5, n = 0.75, etc. What does the total graph look like now? If you extrapolate this out to infinity what would the graph for all \(\displaystyle 0 \le n \le 4\) look like?
How does that help you calculate the integral of this function? As a hint, you may want to review your notes and/or textbook for information about the integral between two curves. If you are having trouble locating it in your notes/textbook, you might try this page from Lamar University.
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Start with the algebra you've learned:
When 0 < x < 1, what is the value of f(x)?
When 1 < x < 2, what is the value of f(x)?
When 2 < x < 3, what is the value of f(x)?
When 3 < x < 4, what is the value of f(x)?
When x = 4, what is the value of f(x)?
How does the above graph?
Using the "area under the line" definition of the integral, drawing the four rectangles, and adding their areas, what value do you get?
Steven G
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ks, Am I missing something or is this function defined for all reals? Please respond. Thanks!
mmm4444bot
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Yeah, I sure did. My mistake.
mmm4444bot
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No biggie. We can constrain the domain of f, as you did, because the integral's bounds are 0 and 4.
I hope lander realizes that only three small rectangles need to be drawn, to find the area.
PS: You may want to consider editing your post (for future readers of the thread). :cool: