Question

Sketch the regions enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. (Do this on paper. Your instructor may ask you to turn in this graph.)
y=sqrt(x)
y=(1/3)x
x=16
i did this problem a long time ago, im currently studying for a math final i have tomorrow. how would you set up the integral?

Answer 1

Split into two integrals
\[\int_0^{9}\sqrt{x}-\frac{x}{3}\, dx+\int_9^{16}\frac{x}{3}-\sqrt{x}\, dx\]

Answer 2

how do you know which boundaries to use for which integral though?

Answer 3

Here : http://www.wolframalpha.com/input/?i=plot+y%3Dsqrt%28x%29%2C+y%3Dx%2F3+x%3D0..16

Answer 4

The 9 is obtained by solving \(\sqrt{x}=\frac{x}{3}\)
To review about this material, watch my video here:
http://www.youtube.com/watch?v=x-dEesoEqpQ

Answer 5

yeah i understand to find the boundaries, we set the two functions equal to eachother and solve. i got 9 and 0 to be the two boundaries, i just didnt know where the x=16 that was given would fit into the integral

Answer 6

I would take a second look at this. It is important to hand draw these, label and everything to get a good feel. sq rt x and 1/3 x create an area but that is a smoke screen for your instructor to take off points. The loop where the three of them meet, apparently 9 to 16 is the area you want.

Answer 7

The 16 of course is given in your problem.

Answer 8

so there would only be one integral, not two ?

Answer 9

Once you find that area (there is only one area) but the question is hinting to you to integrate along y, If you integrate along x, you would need to do two integrations because of how sq rt x and 1/3 x meet.

Answer 10

ohhh okay i gotcha. im going to try to integrate with respect to y and see how that goes. thank you so much :D

Answer 11

Are you serious chaguanas? integrating with respect to y? I am curios how would you set up that integral.

Answer 12

Well, after looking at it, this can be done with respect to x and it is the second part of integral above. But the question is unusually wordy and led me to believe it was a trick question. Usually the question is simply stated find the area bounded by the curves. I hope Anna sees this before it is too late.

Answer 13

Well if you integrate along the \(y\) you will need 3 integrals!

Answer 14

Ha, ha. God help Anna.