## Andresfon12 Group Title find the volume of the given solid: under the surface z=2x+y^2 and above the triangle vertices (1,1),(4,1) and (1,2). one year ago one year ago

1. Andresfon12 Group Title

|dw:1360708771803:dw|

2. TuringTest Group Title

what is the equation of this part of the area?|dw:1360709404014:dw|

3. Andresfon12 Group Title

y=2x?

4. TuringTest Group Title

no,you can see that the slope from x=1 to x=4 is down for that part of the graph, so the coefficient of x will be negative you should use point-slope form way back from algebra I to do this|dw:1360709566120:dw|

5. Andresfon12 Group Title

@TuringTest yes i can see that

6. Andresfon12 Group Title

so -1/3?

7. TuringTest Group Title

yes, so now use point-slope form on either point $$(x_0,y_0)$$ to get the equation of the line $y-y_0=-\frac13(x-x_0)$

8. Andresfon12 Group Title

y-1 = -1/3 (x-1)?

9. TuringTest Group Title

no, because (1,1) is not one of the points on the diagonal line

10. Andresfon12 Group Title

ok

11. Andresfon12 Group Title

the only one that i can use the (1,2) and (1,4)

12. TuringTest Group Title

correct, we are using point-slope form on the diagonal line, so we can only get the equation from points that lie $$on$$ the line.

13. Andresfon12 Group Title

y-2= -1/3 (x-1)

14. TuringTest Group Title

yes, now solve for y...

15. Andresfon12 Group Title

y= -1/3x+1/3 (+2) y=-1/3 x+7/3

16. TuringTest Group Title

yes, and now you need the line that bounds the region below:|dw:1360710827125:dw|

17. Andresfon12 Group Title

same using slope

18. TuringTest Group Title

er, yeah you could, but you should really be able to eyeball this one. Note here that all we have been doing so far is basic algebra to find the equations of the bounds of the region. Just because you are in calc2 does not mean you can forget the basics!! On the contrary, here is where you actually need them most.

19. Andresfon12 Group Title

really is calc 3

20. TuringTest Group Title

Well, what some people call cal3 others call calc2. MIT for instance only has 2 basic calc classes: single and multivariable. But enough semantics, what's the equation of the bottom line?

21. Andresfon12 Group Title

@TuringTest is 1?

22. TuringTest Group Title

y=1, yes so y is bound by the two equations we found, so they will be the bounds for the inner integral, which will be with respect to x. what are the bounds on x, which we will use for the outer integral.

23. Andresfon12 Group Title

x=1, 4 and y= 1 , 1/3x+7/3?

24. TuringTest Group Title

yes :)

25. Andresfon12 Group Title

$\int\limits_{1}^{4} \int\limits_{1}^{1/3x+7/3} xy$ dy dx

26. TuringTest Group Title

the integrand is the function for z, so use that, not xy gotta go, happy integrating!

27. Andresfon12 Group Title

@TuringTest thank you so much

28. Andresfon12 Group Title

sorry is not z=2x+y^2 i just mistype the book is really saying xy