Mark44 said:Have you sketched a graph of the region? The surface z = 16 - 4x2 - y2 is a paraboloid with elliptic horizontal cross sections, and with vertex at (0, 0, 16).
Many times the most difficult part of these double and triple integrals is determining the limits of integration, and that is usually much more difficult if you don't have a good idea of exactly what the region you're integrating looks like.
Cyosis said:Are you sure you looked at the region? The region you describe is a box with sides 1, 16 and 1. Where as the region D is, as said before, a paraboloid with an elliptical cross section with a slice taken off parallel to the zy-plane. Let's simplify the problem a little and ignore the plane x=1. Then we're left with a paraboloid with an elliptical cross section.
Now let's replace the function xy by 1 (once we've found the volume of the paraboloid we can just replace 1 by xy and add the x=1 plane to get the correct answer). Now the integral over region D' is the volume of the paraboloid z=16-x^2-y^2 bounded by the coordinate planes.
A few questions about the properties of this region.
1)Does the parabola have a maximum or minimum?
2)What are the coordinates of its maximum/minimum?
3)What kind of shape does the surface of the base of the parabola have? (circle/ellipse)
4)What is the radius or semi major/semi minor axis of this circle/ellipse?
5)Looking at the paraboloid from z=0 and up, the coordinate planes cut the paraboloid in how many pieces? Is it a whole, half or quarter paraboloid?
If you can answer these questions (hopefully they are clear), then you have properly looked at the region. Write down the volume integral for the paraboloid described above, don't bother calculating it. Now add the x=1 plane, how does this change the boundaries? And finally integrate xy over the region D to get the answer you are looking for.
No, you're still thinking boxes. If you slice the paraboloid into horizontal cross sections, each slice is an ellipse, with the ellipses getting smaller as you go up (as z increases). None of your slices is an ellipse with major axis 4 and minor axis 2, since your solid is bounded below by the plane y = 1. Actually, each slice is a quarter of an ellipse, since the solid is bounded by the x-z plane (the plane y = 0) and the y-z plane (the plane x = 0).After looking at the graph more, I see that the base is an ellipse and that the major axis is 4 and minor is 2. This leaves me to believe the bounds for dx would be from -2 to 1 (because of the line x=1 if we do not ignore it), and the bounds for dy would be from 0 to 4 (taking two times the integration for that)? I don't know about dz, however, I really have trouble imagining the third dimension.
A triple integral is a mathematical tool used to calculate the volume of a three-dimensional shape or the total amount of a quantity in a three-dimensional space. It is important because many real-world problems involve three-dimensional objects or quantities, and triple integrals allow us to solve these problems accurately.
To find bounds for a triple integral, you first need to determine the limits for each variable (x, y, z) based on the given shape or region. This can be done by carefully visualizing the shape and identifying the boundaries in each direction. Once the limits for each variable are determined, they can be plugged into the triple integral formula to find the bounds.
In Cartesian coordinates, the three variables (x, y, z) represent the position of a point in a three-dimensional space. In cylindrical coordinates, the variables (r, θ, z) represent the distance from the origin, the angle from the positive x-axis, and the height of a point in a cylindrical coordinate system. When using triple integrals, the bounds and calculations will differ depending on which coordinate system is used.
Changing the order of integration in a triple integral means rearranging the order in which the integrals are being evaluated. This is useful when the original order of integration results in a complicated or difficult integral. By changing the order, we can simplify the integral and make it easier to solve.
Triple integrals have many applications in fields such as physics, engineering, and economics. They can be used to calculate the mass, center of mass, and moment of inertia of a three-dimensional object. They can also be used to find the volume of a solid, the amount of fluid flowing through a region, or the probability of an event occurring in a three-dimensional space.