## anonymous 5 years ago 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?

1. watchmath

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

2. anonymous

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

3. watchmath
4. watchmath

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

5. anonymous

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

6. anonymous

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.

7. anonymous

The 16 of course is given in your problem.

8. anonymous

so there would only be one integral, not two ?

9. anonymous

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.

10. anonymous

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

11. watchmath

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

12. anonymous

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.

13. watchmath

Well if you integrate along the $$y$$ you will need 3 integrals!

14. anonymous

Ha, ha. God help Anna.