- #1
wubie
Hello,
I am having trouble setting up triple integrals to find a volume of a given solid. Here is one of the questions with which I am having trouble.
Now I can see that the projection of the solid on the xy plane is the circle x^2 + y^2 = 9. And I think I can visualize the plane z = y + 3 with respect to the cylinder - it slices the cylinder in half diagonally. But I am not sure how to set up the triple integral.
This is the way I would set up the integral
Let E be the solid in question. Then
E = {(x,y,z) | -3 <= x <= 3, -(9-x^2)^1/2 <= y <= (9-x^2)^1/2, 0 <= z <= y + 3 }
If I was to integrate using these limits I would then integrate with respect to z first, then y, then x.
Now I know that it would be easier to eventually convert to polar coordinates but I would like to know if the way I set up the triple integral to find the volume of the solid is correct so far.
int.[-3,3] int.[-(9-x^2)^1/2,(9-x^2)^1/2] int.[0,y+3] 1*dz*dy*dx
How does that look so far?
I am having trouble setting up triple integrals to find a volume of a given solid. Here is one of the questions with which I am having trouble.
Find the volume of the region inside the cylinder x^2 + y^2 = 9, lying above the xy plane, and below the plane z = y + 3.
Now I can see that the projection of the solid on the xy plane is the circle x^2 + y^2 = 9. And I think I can visualize the plane z = y + 3 with respect to the cylinder - it slices the cylinder in half diagonally. But I am not sure how to set up the triple integral.
This is the way I would set up the integral
Let E be the solid in question. Then
E = {(x,y,z) | -3 <= x <= 3, -(9-x^2)^1/2 <= y <= (9-x^2)^1/2, 0 <= z <= y + 3 }
If I was to integrate using these limits I would then integrate with respect to z first, then y, then x.
Now I know that it would be easier to eventually convert to polar coordinates but I would like to know if the way I set up the triple integral to find the volume of the solid is correct so far.
int.[-3,3] int.[-(9-x^2)^1/2,(9-x^2)^1/2] int.[0,y+3] 1*dz*dy*dx
How does that look so far?