They might, but then that assumes that taking derivatives is easier than taking integrals. I agree that for most this is true. At least, though, the test would assess their ability to find derivatives. I'm not sure that approximating would help with most of the definite integral problems. I suspect that the integral regions are sufficiently complex, and the anwers sufficiently close, that it would be difficult, unless you had a graping calculator (which I'm sure are not allowed). For example, on #8, with a basic calculator (if it's allowed), you could figure that the graph of integrand is almost linear on (0, 0.3), then it starts to level off, and reaches a max at at x=1 (where f(x) =0.5). So the area is something greater than one half of the rectangle [0, 1]x[0, 0.5]. So the area must be less than 0.5 and greater than 0.25. This rules out maybe all but answer E? But again I need a calculator to check some of the answers. If it's not allowed, then I'm sunk, and it's back to actually finding the closed form solution (trig substitution? I forget.)
How do you assess knowledge of integral calculus on a multiple-choice exam? If you ask for the indefinite integral of something and give four possible expressions, won't the examinee find it easier to differentiate the four expressions? On the Math GRE ftp://ftp.ets.org/pub/gre/Math.pdf they seem to try to get round this by asking for definite integrals, but then can't the examinee use an approximation method?
Forgive me, Mark, I am not too keen on calculus at this time (almost finished with precalculus and will be taking calculus in the next month). However, it seems similar to assessing algebra in the College Algebra CLEP. There are a lot of questions you can simply "plug" in the possible answers into x and if you get a correct equation, BAM! one question closer to passing w/o really knowing how to figure it out (eg. factoring). Best Regards, Bill
