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It's time to write exams, and when writing in committee we often discover differences in usage between various instructors. Here's an example I noticed today.
What is the proper verb to use in a question of the form: "____ the following integrals"? I've seen evaluate, compute, solve, and others.
I'm no native English speaker, but you can tackle that question from the mathematical point of view as well.
The best verb depends on how you view the nature of definite and indefinite integrals.
Indefinite integrals are operators mapping functions to a set of functions or function (as representative of the equivalence class).
Definite integrals are operators mapping functions to numbers for given intervalls or operators mapping numbers (as one boundary) to numbers for given function and other boundary. In that case, it's numbervalued function of numbers.
Consequently, you should use evaluate.
The theory "calculus" is a special calculus. Consequently, you should use calculate, even for indefinite integrals.
If you see integrals as problems, which non-computable integrals or integrals difficult to compute are indeed, then you should use find.
Regard the differential equation $\frac{\mathrm{d}F(x)}{\mathrm{d}x}=f(x)$ to be solved for $F(x)$. Its solution is $F(x)=\int f(x)\,\mathrm{d}x$ (or $\{F(x)\}=\int f(x)\,\mathrm{d}x$).
If you regard the integral as that solution, you should use the verb associated to solutions of algebraic equations: find.
If you regard the integral as the equation itself, you should use the verb associated to (algebraic) equations: solve.
to map. If it should be to instead of onto, I'll change it. 2. Take the example $\int\frac{\sin x}{x}\mathrm{d}x$ for someone who doesn't know the sine integral function yet. It's not a simple calculation/operation to determine the integral, as all usual approaches fail. So, this is a real problem, whose solution needs to be found. The less you know about integrals, the more integrands are problematic. So, one could think, that integrals are problematic in general, with many of them already solved.
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I would avoid the verb solve as I reserve this for things like equations, inequalities and problems. An integral is equal to a number or a function, so verbs like find, evaluate etc are more appropriate.
I'd use compute only for numerical integration methods.
evaluate and find are the two verbs that are used in textbooks and exams that I've come across. I think evaluate is better for a definite integral because the result is a number, and find is OK for both, eg
Find $\int \sin x e^{\cos x}dx$
or
Find the value of $\int _1^\infty \frac 1 {x^2} dx$
or
Evaluate $\int _2^5 x(2x+1)\sin x dx$
I usually phrase it as "Determine $\int x^2\ dx$" or "Determine $\int_1^3 x^2\ dx$". This way
1) it doesn't tip them off to what type of answer they should arrive at, and
2) it allows them to read the symbol $\int$ as either "integral" or (better in my opinion) "antiderivative".
For completeness, in some problems I write "Set-up, but do not evaluate, the integral that computes some geometrical/physical thing." But when I actually want them to perform the integration (which is another phrase I use colloquially but never as instructions) and arrive at an answer, I write "Use integration to find the geometrical/physical thing."
determine is completely general. It doesn't say anything about the nature of the integral and doesn't fix any way of solution. In that case, you need to accept, if students guess the integral somehow and proof it by its derivative.
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The proper verb is whatever verb your textbook uses.
This helped me one time when I wrote "Evaluate this integral", and one student approximated the integral using rectangles. I was able to show the student that every problem I assigned from the textbook that required a symbolic answer said "Evaluate this integral" whereas every problem that required a numerical approximation said "Approximate this integral". The student did not argue.
This is a good policy for any exam problem, not just integrals. I always use the same phrasing on exams that my textbook uses.
From a general test writing standpoint: If you and the students both know that the problems are to be solved by integrating a formula, why not "Integrate the following..." If they have to figure out whether to use integration or some other calculus then that won't work.
In practical applications, such as HPLC analysis, the usual instruction is calculate the amount of substance X by integrating the peak area in the chromatogram in comparison to a standard chromatogram
