Integral with functions of two variablesHelp

[christian0710]

I really need help with this problem.

Find the volume of the solid enclosed by the surface z=1+e^xsiny and the planes x=±1, y=0, y=pi and z= 0


My logic tells me that
we have some rectangles R[-1,1]x[0,pi] but I'm confused by the z=0

 

[LCKurtz]

I really need help with this problem.

Find the volume of the solid enclosed by the surface z=1+e^xsiny and the planes x=±1, y=0, y=pi and z= 0 My logic tells me that
we have some rectangles R[-1,1]x[0,pi] but I'm confused by the z=0

The planes $x=\pm 1, y=0, y=\pi$ form the vertical side walls of the solid. The plane $z=0$ forms the "floor" or base of the solid. The equation $z=e^x\sin y+1$ is the curvy roof of the solid. Does that help?

 

[christian0710]

This gives me a better picture of what is going on :)
So I will double integrate the function z, first with respect to x in the inverval [-1,1] and then with respect to y in the interval [0,2]

I guess there is no reason to include z=0, or do i somehow have to include that plane in the equation as well?

 

[LCKurtz]

This gives me a better picture of what is going on :)
So I will double integrate the function z, first with respect to x in the inverval [-1,1] and then with respect to y in the interval [0,2]

I guess there is no reason to include z=0, or do i somehow have to include that plane in the equation as well?

It is automatically included in that formula for volume:$$
\iint_A (z_{upper} - z_{lower})\, dydx$$In this case $z_{lower}= 0$.

 

[christian0710]

Ahh i see. so upper is z=e^x*sin(y) +1 and if we integrate it with respect to x we get e^x*sin(y)+x because y is a constant so y is a constant right? :) and then we insert upper and lower limit (-1,1) and do the same for y with limits [0,2]

 

[LCKurtz]

Ahh i see. so upper is z=e^x*sin(y) +1 and if we integrate it with respect to x we get e^x*sin(y)+x because y is a constant so y is a constant right? :) and then we insert upper and lower limit (-1,1) and do the same for y with limits [0,2]

Yes.

 

[christian0710]

When i integrate with respect to x first with limit [-1,1] i get e*sin(y)-e^-1*sin(y)

Jesus i got to lern to write in latex! '
if i type ∫ e*sin(y)-e^-1*sin(y), y,0,2) into my calculator i get 3,32 does that sound right. And do you know of any way to type the whole ting into a ti89 with limits for both?

Btw. Thank you for helping out!

 

[LCKurtz]

When i integrate with respect to x first with limit [-1,1] i get e*sin(y)-e^-1*sin(y)

Jesus i got to lern to write in latex! '
if i type ∫ e*sin(y)-e^-1*sin(y), y,0,2) into my calculator i get 3,32 does that sound right. And do you know of any way to type the whole ting into a ti89 with limits for both?

Btw. Thank you for helping out!

That looks correct, but why would you want to use a calculator in the first place? Can't you just leave the exact answer in terms of $e$ and $\cos 2$?

 

[christian0710]

Yes I can and finally did it all by hand!
It always feels rewarding when you succeed in doing the things you find most difficult :)