Problem 15 - Advanced Engineering Mathematics

Question

In Problems \(15-20\), sketch the region \(D\) whose volume \(V\) is given by the iterated integral. $$ \int_{0}^{4} \int_{0}^{3} \int_{0}^{2-2 / 3} d x d z d y $$

Step-by-Step Solution

Answer

The region is a rectangular prism in the first octant bounded by \(0 \leq x \leq \frac{2}{3}\), \(0 \leq z \leq 3\), \(0 \leq y \leq 4\).

Step 1: Understand the integral limits

The iterated integral is given as \( \int_{0}^{4} \int_{0}^{3} \int_{0}^{2-2/3} dx \, dz \, dy \). Each integral has limits that define the range for the respective variable. Here, \(x\) ranges from \(0\) to \(\frac{2}{3}\), \(z\) ranges from \(0\) to \(3\), and \(y\) ranges from \(0\) to \(4\). These limits describe a region in three-dimensional space.

Step 2: Analyze the region for x

The innermost integral \(\int_{0}^{2-2/3} dx\) suggests that for every fixed \(z\) and \(y\), the variable \(x\) ranges from \(0\) to a constant value \(\frac{2}{3}\). This describes a slice parallel to the yz-plane.

Step 3: Analyze the region for z

The next integral \(\int_{0}^{3} dz\) indicates that for each fixed \(y\), \(z\) ranges from \(0\) to \(3\). Thus, the region for \(z\) is a layer extending from the \(xz\)-plane, entirely spanning from \(0\) to \(3\) in the positive \(z\)-direction.

Step 4: Analyze the region for y

The outermost integral \(\int_{0}^{4} dy\) shows that the variable \(y\) varies from \(0\) to \(4\). This means that the entire structure, defined by the previous limits for \(x\) and \(z\), is extruded in the \(y\)-direction from \(0\) to \(4\).

Step 5: Sketch the region

Combine the information from the previous steps to sketch the region \(D\). The region is a box or rectangular prism in the first octant bounded by \(0 \leq x \leq \frac{2}{3}\), \(0 \leq z \leq 3\), and \(0 \leq y \leq 4\). It is a finite, solid region in space.

Key Concepts

Sketching Regions

When working with iterated integrals, sketching the corresponding region is a useful skill that helps visualize the area being integrated. In this exercise, we want to identify and sketch the region defined by the integral: \[\int_{0}^{4} \int_{0}^{3} \int_{0}^{\frac{2}{3}} dx \, dz \, dy\] By looking at the limits and their respective integrals, you can determine how the region unfolds in three-dimensional space.

• The innermost limit of \(x\) goes from \(0\) to \(\frac{2}{3}\).
• The middle limit of \(z\) spans from \(0\) to \(3\).
• The outermost limit for \(y\) ranges from \(0\) to \(4\).

Once we identify these limits, sketching the region becomes easier. Imagine stacking layers or building blocks from the baseline sizes defined by these ranges. For this problem, the region looks like a block or rectangular prism in the positive octant.

Three-Dimensional Space

Three-dimensional space is an environment where each point is determined by a coordinate triplet \((x, y, z)\). Understanding iterated integrals in this space involves visualizing how these variables interact to form a solid. In our example, \(x\), \(y\), and \(z\) each define orthogonal dimensions. The limits on these variables shape the physical boundaries of the region.

• \(x\): Variable that moves within the plane defined by fixed \(y\) and \(z\).
• \(z\): Defines how far the slice increases into depth, towards the front and back from one end to another.
• \(y\): The direction of extrusion, moving the slices and layers upward.

When such volumes are visualized, each section defined by the upper and lower bounds appears as a bounding box limiting or defining part of the space. These bounding boxes combine to form the complete region of integration.

Limits of Integration

The limits of integration in an iterated integral incrementally define the region over which we integrate, one dimension at a time. In three-dimensional space, these help sculpt the volume enclosed by the integral. For our integral: \[\int_{0}^{4} \int_{0}^{3} \int_{0}^{\frac{2}{3}} dx \, dz \, dy\] Each limit determines the range of values for a particular variable:

• From 0 to \(\frac{2}{3}\) for \(x\) signifies that along the x-axis, our region is clipped between these two boundaries.
• From 0 to 3 for \(z\) means the structure grows lengthwise between the front and back faces.
• From 0 to 4 for \(y\) sets the height, stretching the volume upwards.

By meticulously following each limit, you can accurately imagine or sketch the cubic region that characterizes the bounds of the integral. Such comprehensive understanding is crucial for effectively solving iterated integrals.