Question

find the volume of the given solid over the indicated region of the integration.
f(x,y) = 2x+3y+2; -2<=x<=4; 2<=y<=5

Answer 1

This is how I've started
\[\int\limits_{0}^{4} \int\limits_{2}^{5}2x +3y +2dydx -\int\limits_{-2}^{0}\int\limits_{2}^{5}2x +3y +2dydx\]

Answer 2

Solved for the inside integral of both first
\[\int\limits_{2}^{5}(2x +3y +2) dy = 2xy + \frac{ 3 }{ 2 }y^2 +2y |\]
\[ y=5: 10x+\frac{ 75 }{ 2 }+10\]
\[-(y=2:4x+48+4)\]
\[= 6x- 9/2\]

Answer 3

Am I on the right track?

Answer 4

why take a negative sign?
it can be -2 to 4.
\[V=\int\limits_{-2}^{4}\left( 6x-\frac{ 9 }{ 2 } \right)dx\]

Answer 5

Finding volume. Wouldn't the negative part be subtracted from the total that way?

Answer 6

i don't think so.

Answer 7

\[V=\left[ \frac{ 6 x^2 }{ 2 }-\frac{ 9 }{ 2 }x \right]~from~-2~\to~4=3\left( 4^2-(-2)^2 \right)-\frac{ 9 }{ 2}\left( 4-(-2) \right)\]

Answer 8

wolfram alpha says 261, this says 9, I found 43 my way (which was wrong)

Answer 9

your splitting doesn't look correct

Answer 10

the solid is living both above and below xy plane and you're spluttung because you want to integrate the absolute value, right ?

Answer 11

right

Answer 12

can u give me wolfram link

Answer 13

http://www.wolframalpha.com/input/?i=integral+of+2x%2B3y%2B2+%3B+-2%3C%3Dx%3C%3D4+%3B+2%3C%3Dy%3C%3D5

Answer 14

wolfram is giving you signed volume

Answer 15

you should get same or more than that if u take absolute value

Answer 16

http://www.wolframalpha.com/input/?i=integral+of+abs%282x%2B3y%2B2%29+%3B+-2%3C%3Dx%3C%3D4+%3B+2%3C%3Dy%3C%3D5

Answer 17

Same result, then?

Answer 18

Yes because the solid is completely above the xy plane in the given region

Answer 19

no splitting is needed

Answer 20

Do you know the correct way to set this up?

Answer 21

`f(x,y) = 2x+3y+2; -2<=x<=4; 2<=y<=5`

first notice that

\[\large f(x,y)= 2x+3y+2 \ge 0 \\~\\ \text{ when } -2\le x\le 4 \text{ and } 2\le y \le 5 \]

Answer 22

So

\[\int\limits_{-2}^{4} \int\limits_{2}^{5}\Bigg|2x +3y +2\Bigg|dydx =\int\limits_{-2}^{4} \int\limits_{2}^{5}2x +3y +2~dydx \]

Answer 23

http://gyazo.com/8b19ba6aa78f053b44006987b0a9c7bd

Answer 24

leads to this?\[\int\limits_{-2}^{4} \left( 2xy +\frac{ 3 }{ 2 } y^2 + 2y from 2\rightarrow5\right)dx\]

Answer 25

looks good

Answer 26

\[\int\limits_{-2}^{4}6x-\frac{ 9 }{ 2 }dx\]

Answer 27

im getting 6x + 75/2

Answer 28

\[\int\limits_{-2}^{4}6x+\frac{ 75 }{ 2 }dx\]
\[\left( 3x^2 +\frac{ 75 }{ 2 }x \right)\]
from -2 to 4

Answer 29

Yep

Answer 30

=261

Answer 31

!! Thank you again ganeshie!

Answer 32

np :)