91Ó°ÊÓ

A triple integral is a mathematical tool used to calculate the volume of a three-dimensional region. You can think of it as an extension of double integrals, which are used to find the area under a curve in a 2D space. In the context of computing volume, a triple integral sums up infinitely small volumes in a three-dimensional object. By integrating over three variables usually denoted by \( x, y, \text{ and } z \), we obtain the total volume.The general form of a triple integral is:

• \( \int \int \int f(x, y, z) \, dz \, dy \, dx \).
• Here, \( f(x, y, z) \) is the function representing the density at each point, but often it is simply \( 1 \) when finding the volume.
• The limits of integration correspond to the boundary constraints of the region.

In this exercise, the expression \( \int_{0}^{5} \int_{0}^{10-2x} \int_{0}^{10-2x-y} 1 \, dz \, dy \, dx \) was used to calculate the volume of a solid. This integral is evaluated in steps, first with respect to \( z \), then \( y \), and finally \( x \). First integrating \( z \) with limits \( 0 \leq z \leq 10-2x-y \), and successively with \( y \) and \( x \).

The first octant is a crucial concept in three-dimensional geometry. It pertains to the portion of space where all three coordinates \( x, y, \text{ and } z \) are positive. Visualize a 3D Cartesian coordinate system divided by three mutually perpendicular planes—the \( xy \)-plane, \( yz \)-plane, and \( zx \)-plane—intersecting each at the origin. These planes divide space into eight sections or 'octants'.

• The first octant is found in the positive direction of all three axes: \( x > 0, y > 0, \text{ and } z > 0 \).
• This region is essential for problems involving spatial geometry, as it establishes a zone where all points have non-negative coordinates.

In solid modeling and integration, as seen in the exercise, it means that any point considered lies within this all-positive region. Thus, the first octant simplifies calculations as it limits evaluations to non-negative coordinate values, guiding the boundaries, such as \( 0 \leq x \leq 5 \) based on where the plane intersects the axes.

A plane equation in three dimensions is typically expressed as \( z = ax + by + c \), which describes a flat surface in space. Finding where a plane intersects the axes provides critical information about a region's boundaries and constraints. Understanding these intersection points helps to establish limits for integration.

• For the \( x\)-axis, set \( y = 0 \) and \( z = 0 \). Solve the equation to find where the plane crosses.
• For the \( y\)-axis, set \( x = 0 \) and \( z = 0 \), then solve for \( y \).
• For the \( z\)-axis, set \( x = 0 \) and \( y = 0 \). This directly gives the \( z \) intersection.

In the given problem, the plane \( z = 10 - 2x - y \) intersects the axes at:

• Point \( (5, 0, 0) \) on the \( x\)-axis,
• Point \( (0, 10, 0) \) on the \( y\)-axis, and
• Point \( (0, 0, 10) \) on the \( z\)-axis.

These intersection points effectively limit the region over which the volume is calculated by aiding in the selection of integration limits.