Problem 10 - Thomas Calculus

Question

Integrate \(G(x, y, z)=y+z\) over the surface of the wedge in the first octant bounded by the coordinate planes and the planes \(x=2\) and \(y+z=1 .\)

Step-by-Step Solution

Answer

The integral of \(G(x, y, z) = y + z\) over the surface is 2.

Step 1: Understand the Surface

The surface is bounded by the coordinate planes, the planes \(x=2\) and \(y+z=1\) in the first octant. This means our region of integration will be confined to where \(x\), \(y\), \(z\) are non-negative, \(x\leq 2\), and \(y+z\leq 1\).

Step 2: Determine the Limits of Integration

Since we need to integrate over the surface defined by \(y + z = 1\), we will express one variable in terms of the other. Choose \(y\) as independent and let \(z = 1 - y\). The limits for \(y\) will be from \(0\) to \(1\). The \(x\) limits are from \(0\) to \(2\), as defined by the plane \(x = 2\).

Step 3: Set Up the Integral

The integral of \(G(x, y, z) = y + z\) over the surface involves integrating against the area element of the surface in the \(y-z\) plane. Since \(z = 1 - y\), rewrite \(G(x, y, z)\) as \(y + (1 - y) = 1\). The double integral becomes: \[\int_{x=0}^{2} \int_{y=0}^{1} 1 \, dy \, dx\]

Step 4: Evaluate the Inner Integral

Evaluate the inner integral with respect to \(y\): \[\int_{y=0}^{1} 1 \, dy = [y]_{0}^{1} = 1 - 0 = 1\]

Step 5: Evaluate the Outer Integral

The outer integral with respect to \(x\) becomes: \[\int_{x=0}^{2} 1 \, dx = [x]_{0}^{2} = 2 - 0 = 2\]

Step 6: Final Result

The value of the surface integral over the wedge is 2.

Key Concepts

Coordinate Planes

Coordinate planes in a three-dimensional space are the planes formed by all the combinations of two of the three axes: the x-axis, y-axis, and z-axis. These planes effectively help define the boundaries for regions in space, which is particularly useful when setting up integrals. In this context:

• The XY-plane is where the z-coordinate is zero, making it useful for visualizing surfaces relative to these axes.
• The YZ-plane is set when x is zero, often useful when dealing with planes such as x=constant.
• The XZ-plane is where the y-coordinate is zero, handy when working with planes like y+z = constant.

Understanding these planes helps in tasks such as visualizing the wedge's surface described by the specific planes it is intersecting. For example, in our exercise, the wedge is intersected by the planes x=2 and y+z=1, both of which rely on these foundational coordinate planes for orientation.

Limits of Integration

When evaluating integrals over a specific region, it's crucial to establish the correct limits of integration. They dictate where the region starts and ends, ensuring the correct portion of the graph is considered. For surfaces like our wedge, limits are determined by intersecting planes:

• The limits for x are dictated by the plane x=2, giving a range from 0 to 2.
• For y and z, given by the plane y+z=1, where we chose one as independent (y from 0 to 1), while z can be expressed as z=1-y, keeping simplicity in checking boundaries.

These bounds shape the wedge in the first octant and define each variable's extent during the integration process.

Wedge Surface

A wedge surface in mathematics refers to a three-dimensional corner or slice formed by intersecting planes in a defined sector of space, often bound by coordinate planes. In our exercise, the wedge is bound by:

• The wall x=2, limiting the extent along the x-axis.
• The slope y+z=1 in the first octant, dividing the space diagonally, simplifying equations.

This configuration shapes a right-angled corner, forming part of our solution's integral region. By manipulating variables and understanding boundaries, the integration over this wedge becomes more straightforward, aiding in solving surface integrals efficiently.

Octants in 3D Space

3D coordinates are divided into eight sections called octants, similar to quadrants in 2D space. Each octant is defined by the signs (positive or negative) of x, y, and z coordinates. The first octant, which is our area of interest, is where all three coordinates - x, y, and z - are positive.

• First Octant: (x ≥ 0, y ≥ 0, z ≥ 0) ensures calculations and integrations, like in our example, remain in a straightforward, positive domain.
• This segmentation is crucial in understanding how different planes and surfaces interact within three-dimensional space, informing limits and integration paths appropriately.

By analyzing the octants, especially the first, we ensure our calculations are precise and comprehensive, applying only to the correct geometric area.