I’ve hit a brick wall in my math homework. There’s something obvious that I’ve forgotten, and I’m missing a connection in my brain somewhere.
“Let f(x) = 3- √x. Take the derivative of the function ∫^(2x\^3) ₄ f(t)dt using the Fundamental Theorem of Calculus, Part 1. If necessary, rewrite the integral expression first.”
It’s a multiple choice question and none of the answers include a “t”. Where did it go? What is up with the “t”? Why is it there? I swear I knew this like 3 days ago, but the knowledge left my brain as soon as I stopped looking at that problem, and I didn’t write it down. The textbook does not mention the “t” after stating the example problems, so I know whatever I’m doing to it is probably step 1.
In: 7
T is just a placeholder for time. Did you learn that integrals and double/triple integrals give you acceleration, velocity, etc.?
There is no T in the answer. Because you are plugging in the start/end points of the given definite integral.
The theorem just states that if the function is continuous in a range, then you can pick any start/end point in that range for the integral of that function and its derivative is the original function.
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> Let f(x) = 3- √x. Take the derivative of the function ∫2×3) ₄
I’ll take this as the start point is 4 and the end point is 2x/3.
So take the integral of 3- √x:
f(x) = 3 – x^(0.5)
∫f(x) = F(x) = 3x – (x^(1.5) /1.5 ) + c
Now do F(2x/3) – F(4)
[3(2x/3) – (2x/3)^(1.5) /1.5] – [3(4) – (4)^(1.5) /1.5]
*t* is a “dummy variable” that you are summing over with your *definite* integral.
The reason you need it is because you have an *x* in the limits of the integral, so you need a second variable inside the integral.
An integral is a special kind of sum; an infinite sum of infinitely small things.
Imagine if we had a normal sum. Let’s say we were summing *n^2* from *n = 1* to *n = 5*. We would get 55. It wouldn’t surprise us that there is no *n* in the answer – of course there isn’t, it is just a placeholder we are using as part of the sum.
But now suppose we wanted the sum to an unspecified number of terms. We could write that as the sum of *n^2* from *n = 1* to *n = n*, in which case we would expect to get an *n* in our answer (and we would, *n(n+1)(2n+1)/6*)… but that gets a bit confusing, as we have *n* in two places being two different things. In one place it is being an unspecified number that we want in our answer (standing in for any possible integer), in other places it is being an index, the thing we are summing over (so taking all integer values from 1 up to *n*). So instead we would normally write that using two letters, so the sum of *k^2* from *k = 1* to *k = n*. Now we will get an *n* in our answer, but we are being clear about the differences between the index *k* (or we could use *i*, maybe *m* – the thing taking *all* values in the range), and the unspecified, generalised *n* (which can take *any* value we want).
We tend not to think about this as much with integration, even though it is a type of sum (the ∫ is a curly “s” for “sum”).
When we do *indefinite* integrals we gloss over this; we are effectively doing a definite integral from some unspecified lower bound up to *x* (or whatever our variable is), and we could use a different variable within the integral to be clear, but it doesn’t really matter as we do this all implicitly.
But with a *definite* integral, if we have a variable in the limits of the integral we should be clear that this is a different variable to what is in the integral itself. So in your example *t* is the dummy variable we are integrating over, x is the variable that we want in our limit. Within our integral *t* is taking any value from the lower limit of the integral to the upper limit. If I’m reading this correctly that means *t* is taking values from *4* up to *2x/3*.
It is a notation thing. But it is an important notation thing.
We use *t* because that is often used for a parameter or a dummy variable (due to its use as “time” in physics). We could put anything else in there (other than *x*, as we’ve already used that, or any other letter we’ve also used), following our general rule in maths that we can call things whatever we want. We could use the dummy variable *banana*, replacing all *t*s with *banana*s and it would still work the same.
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