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So I took a guess on this question and got it right but after multiple attempts to understand it, I still don't get why that was the right answer. Could you explain why the following is the case? I attempted to solve the following via integrations but my calculator got overloaded and I had to restart it.
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So here's a rough explanation for how to verify the solution. Solving for it is a little more complicated, which I can show you. But essentially both the bounds of the integrals as well as the integrand depend on k, so it's complicated to solve in a closed form expression to solve for k. It would probably be easier to eyeball where the right solution is, and then compare the multiple choice. For the AP exam, I'd say as long as you can setup the equal integrals, that's all they would expect out of you.
There's actually a much simpler way to solve it. Basically, k is the average value of the function.
Thank you very much! Now that I see it in that sense, it is more enlightening. Once again, thank you!