Question

Find the volume between the surfaces given by: z=f(x,y)=2+2x-4y and z=g(x,y)=3+x^2 over the triangular plane region D with vertices (0,0), (3,0), and (3,6). { double or triple integral}

Answer 1

what have you tried so far

Answer 2

I tried graphing the plane and parabola. I have trouble graphing the plane and setting up the integral. Mainly the boundaries

Answer 3

@ganeshie8

Answer 4

I tried finding the axes for the plane. I got z=2, y=1/2, and for x, i sometimes get 1 and other times i get -1

Answer 5

but the top surface is 2+2z-4y and the bottom is 3+x^2 right?

Answer 6

it looks like this i think :
http://gyazo.com/1f2ba93b9f44e8bae56d6e4d509782cf

Answer 7

ok so thats the graph i got the first time i did it, but i now im confused as to whos the top and whos the bottom

Answer 8

yay ur finally back

Answer 9

i just need clarification on whos the top surface and bottom

Answer 10

hey are u really sure there are no typoes in the question
because we need to setup 3 integrals to find the volume :O

Answer 11

i triple checked. it could be either double or triple cus when we set it up as triple then the function inside is just 1 or a constant.

Answer 12

we need to setup 3 triple integrals

Answer 13

WHAT

Answer 14

not just 1 triple integral, but 3?

Answer 15

depends, if u want signed volume then 1 triple integral will do

Answer 16

lets assume they want the signed volume

Answer 17

ok i think its just 1 triple integral. you got me freaking out

Answer 18

\[\int\limits_0^3\int\limits_0^{2x} ((2+2x-4y) - (3+x^2)) ~dydx\]

?

Answer 19

Ok, so the plane is the top surface and the parabola is the bottom?

Answer 20

how can you tell? thats why im asking

Answer 21

its not same over the entire region,
look at the earlier plot

Answer 22

i saw, but its like when the plane slices the parabola thingy, the values of x are negative, so i was hesistant to do that

Answer 23

i get what you mean, you need to setup 3 triple integrals if you want the actual volume

Answer 24

and what do you mean by not the same, i think our teacher just wants the main one, like the one nearest to the "origin"

Answer 25

if you want signed volume, just take the difference of functions

Answer 26

ahhhh so for the boundaries can u briefly tell me how u got them

Answer 27

look at the given region in xy plane

Answer 28

oh i see the 2x now. its the slope ahhhhhh omg i get this

Answer 29

yes :) do u have the final answer ?

Answer 30

so i get how to set up the integral now, so can you explain to me how to graph the plane. i keep on getting x=1 and x=-1

Answer 31

like when i put x=y=0, x=z=0, and y=z=0

Answer 32

z = 2+2x-4y

find the intercepts

Answer 33

x intercept = -1
y intercept = 1/2
z intercept = 2

yes ?

Answer 34

yes i got that, but when you plug it back those intercepts back into the eqn, they're not equal

Answer 35

2=2+2(-1)-4(1/2)
2=2-2-2
2=/=-2

Answer 36

x intercept = -1, so the point would be `(-1, 0, 0)`
y intercept = 1/2
z intercept = 2

Answer 37

you need to plugin the "point" corresponding to x intercept

Answer 38

do you know how to plot the line `x+y = 1` in xy plane ?

Answer 39

oh...right yeah. thanks for the reminder.

Answer 40

x intercept = 1
y intercept = 1

do u expect that line to satisfy the point (1, 1) ?

Answer 41

thanks for the help. appreciated

Answer 42

np :)