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This was a question that came up on a short pre-selection exam to enroll in a CS Master degree program.

Possible answers were:

  • a) not enough information
  • b) $ 16 $
  • c) $ 31 $
  • d) $ \frac{17}{2} $

I've tried simplifying:

$$ \int_1^2 3f(\frac{x}{2}) + 1\ dx $$

Here's what I've done:

  1. Extract constant and apply integral addition rule.

$$ 3\int_1^2 f(\frac{x}{2}) \ dx + \int_1^2 dx $$

  1. Compute the second integral of the sum.

    $$ 3\int_1^2 f(\frac{x}{2}) \ dx + x\rvert_1^2 = 3\int_1^2 f(\frac{x}{2}) \ dx + 1 $$

I'm not sure how to proceed from here on.

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    $\begingroup$ Are you sure the problem has been copied directly? The first integral involves the values of $f(t)$ when $1<t<2$, while the second integral involves the values of $f(t)$ when $\frac12<t<1$; they have nothing to do with each other. $\endgroup$ Sep 6, 2023 at 3:42
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    $\begingroup$ To paraphrase @GregMartin, you can say nothing about the second integral. $\endgroup$
    – copper.hat
    Sep 6, 2023 at 4:01
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    $\begingroup$ In Case the given Integral had limits $(0,1)$ , then the wanted Integral will have limits $(0,0.5)$ & we still can not say much about that , unless we are given a POSITIVE function , which will then make the wanted Integral between $0$ & $16$. When we are then given monotonic function , we might say a little more. Currently , we can say almost nothing. $\endgroup$
    – Prem
    Sep 6, 2023 at 4:19
  • $\begingroup$ @GregMartin I've double-checked, the problem has been copied correctly. It was a multiple-choice answer task and your reply was one of the offered options. So the substitution rule changes the limits of integration. Hence we're looking at a different region under the curve $ f (t) $ and we can't say anything about the second integral in relation to the first one. Greg Martin, would you mind promoting your comment to an answer? $\endgroup$ Sep 6, 2023 at 10:05
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    $\begingroup$ a) is the correct answer as there is no information concerning $f$ on $[1/2,1).$ $\endgroup$ Sep 6, 2023 at 18:08

1 Answer 1

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The given equation gives us the area under $f(x)$ between the lines $x = 1$ and $x=2$ However, when you look at the second expression, you are asked about the area which is derived from the area under $f(x)$ between the lines $x = \frac{1}{2}$ and $x = 1$ .

now since the function $f$ is not defined to us, we can assume any $f$ as long as it satisfies the first condition. But 2 different choices of $f$ may not give us the same value for the second expression as it covers a different area. Thus we can infer that we have not been given enough information. We can demonstrate this with an example.

  • first choice of $f$: $f(x)=5$

Clearly$\int_1^2 f(x)dx=5$, so this equation is valid. according to this choice of function $f$, the value of the second expression would be (as you worked out yourself): $$ 3\times 2.5 +1 = 8.5$$

  • second choice of $f$: $f(x) = \frac{10}{3}x$

again $\int_1^2 f(x)dx=\left[ \frac53 x^2\right]_1^2 = \frac{20}3 - \frac53=5$ so this is also valid. according to this, your second expression becomes: $$3\times \left[ \frac53 x^2 \right]_\frac12^1+1$$ $$=3\times \left[ \frac53 - \frac5{12} \right]+1$$ $$=3\times \left[ \frac{15}{12} \right]+1$$ $$=\frac{15}4 +1$$ $$=\frac{19}4$$ $$=4.75$$

As the values of the expressions are different, option $a$ would be correct.

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