Problem 18 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 11-20, sketch the solid \(S\). Then write an iterated integral for $$ \iiint_{S} f(x, y, z) d V $$ \(S\) is the region in the first octant bounded by the surface \(z=9-x^{2}-y^{2}\) and the coordinate planes.

Step-by-Step Solution

Answer

The iterated integral for the region is \[\int_{0}^{3} \int_{0}^{\sqrt{9-x^2}} \int_{0}^{9-x^2-y^2} f(x, y, z)\, dz\, dy\, dx\].

Step 1: Identify the Boundaries of the Region

The region \( S \) is in the first octant, meaning \( x, y, \) and \( z \) are all non-negative. Additionally, \( z \) is capped by the surface \( z = 9 - x^2 - y^2 \). The boundary condition is thus given by the inequality \( z \geq 0 \). This implies that \( 9 - x^2 - y^2 \geq 0 \), which simplifies to \( x^2 + y^2 \leq 9 \). Hence, the region in the \( xy \)-plane is a circle with radius 3, centered at the origin.

Step 2: Set Up the Iterated Integral Limits

To write the iterated integral, we need to establish limits for \( z \), \( x \), and \( y \). The variable \( z \) ranges from 0 to \( 9 - x^2 - y^2 \). Variable \( y \), for constant \( x \), will range from 0 to the semicircle boundary in the first octant, i.e., \( 0 \) to \( \sqrt{9 - x^2} \). Finally, the variable \( x \) ranges from 0 to 3.

Step 3: Write the Iterated Integral

Given the limits identified, the iterated integral over the region \( S \) is set up as follows: \(\int_{0}^{3} \int_{0}^{\sqrt{9-x^2}} \int_{0}^{9-x^2-y^2} f(x, y, z)\, dz\, dy\, dx\). Here, the order of integration chosen respects the dependencies of each variable, with \( z \) being most dependent.

Key Concepts

Triple Integral

In calculus, a triple integral is a way to integrate over a three-dimensional region. Just like a single or double integral allows us to calculate values like area or volume in one or two dimensions, triple integral lets us operate in three dimensions. When we talk about triple integrals, we refer to the collective representation of a function over a three-dimensional space called a solid region, which is denoted as \( \iiint_{S} f(x, y, z) \, dV \). Here, function \( f(x, y, z) \) could represent density or temperature, and \( dV \) represents a small element of volume. The triple integral sums up values of \( f(x, y, z) \) over all these small volume elements inside \( S \). Short and repetitive practice is best to fully understand triple integrals.

First Octant

The first octant is a concept specific to three-dimensional geometry, specifically in the coordinate system. We split 3D space into eight sections known as octants. The first octant is the particular section where all the coordinates \( x, y, \) and \( z \) are non-negative. Think of it more plainly: It’s like the top-front corner of a 3D graph where values rise from zero upwards along all three axes. In our exercise, when we say the integral covers the first octant, we are considering only those parts of the solid region \( S \) that lie within this non-negative section. This simplifies the integration process because it limits the area we need to think about.

Bounded Region

A bounded region in three-dimensional space is confined within certain limits or boundaries. Understanding the bounded region is crucial when solving problems involving integrals since it tells us where to perform our calculations.In the given exercise, the region \( S \) is bounded by a surface described by the equation \( z = 9 - x^2 - y^2 \). Additionally, it is restricted to the first octant, meaning any values of \( x, y, \) and \( z \) can only be positive or zero.By using the equation \( z = 9 - x^2 - y^2 \), we conclude that \( x^2 + y^2 \leq 9 \), indicating the region is shaped by a circular boundary with radius 3 on the \(xy\)-plane. Recognizing this helps in correctly setting integration limits and envisioning the spatial region encompassed by \( S \). Also, it reinforces us on how to limit values for each variable, such as letting \( x \) go from 0 to 3, where the circle meets the axes.

Coordinate Planes

Coordinate planes in three-dimensional space are the flat, two-dimensional stretches where two out of the three variables (coordinate axes) lie flat, while the third variable remains constant. We have three primary coordinate planes in 3D space:

• The \( xy \)-plane where \( z = 0 \)
• The \( xz \)-plane where \( y = 0 \)
• The \( yz \)-plane where \( x = 0 \)

In the context of our problem, these coordinate planes act as boundaries for the region \( S \). For example, the constraint \( z \geq 0 \) is a restriction brought forth by the coordinate plane \( xy \). The boundaries defined by these planes ensure that our region lies purely in the first octant. By understanding that these planes represent intersecting surfaces within our 3D space, it simplifies our visualization of the space where our integration problem lies and aids in determining the integration limits.