Problem 17 - Thomas Calculus

Question

Write an iterated integral for \(\iint_{R} d A\) over the described region \(R\) using (a) vertical cross-sections, (b) horizontal cross-sections. \begin{equation}\begin{array}{l}{\text { Bounded by } y=3-2 x, y=x, \text { and } x=0} \end{array} \end{equation}

Step-by-Step Solution

Answer

Vertical: \( \int_{0}^{1} \int_{x}^{3-2x} \, dy \, dx \). Horizontal: \( \int_{0}^{1} \int_{0}^{\frac{3-y}{2}} \, dx \, dy \).

Step 1: Understand the Region

The region \( R \) is bounded by the lines \( y = 3 - 2x \), \( y = x \), and \( x = 0 \). Visualize the region on the Cartesian plane. The line \( y = 3 - 2x \) is a downward sloping line with a y-intercept of 3, and \( y = x \) is a diagonal line at 45 degrees through the origin. The boundary \( x = 0 \) is the y-axis.

Step 2(a): Determine Limits for Vertical Cross-Sections

For vertical cross-sections, we integrate with respect to \( y \) first and \( x \) second. Observe that for a given \( x \), \( y \) varies from the line \( y = x \) to the line \( y = 3 - 2x \). The limits for \( y \) are:\[ x \leq y \leq 3 - 2x \]Next, determine the range for \( x \) by finding the intersection points of \( y = 3 - 2x \) and \( y = x \). Solving these gives \( x = 1 \). Therefore, \( x \) ranges from 0 to 1.

Step 3(a): Write the Iterated Integral (Vertical)

With limits determined, the iterated integral using vertical cross-sections is:\[ \int_{0}^{1} \int_{x}^{3-2x} \, dy \, dx \]

Step 2(b): Determine Limits for Horizontal Cross-Sections

For horizontal cross-sections, integrate with respect to \( x \) first and \( y \) second. Here, \( y \) varies from 0 to the intersection of \( y = 3 - 2x \) and \( y = x \) which is at \( y = 1 \). For each \( y \), \( x \) ranges from \( x = 0 \) (the y-axis) to the x-values on the lines \( y = x \) (which is \( x = y \)) and \( y = 3 - 2x \) (solve for \( x = \frac{3-y}{2} \)). The limits for \( x \) are:\[ 0 \leq x \leq \frac{3-y}{2} \] for \( 0 \leq y \leq 1 \).

Step 3(b): Write the Iterated Integral (Horizontal)

With the determined limits, the iterated integral using horizontal cross-sections is:\[ \int_{0}^{1} \int_{0}^{\frac{3-y}{2}} \, dx \, dy \]

Key Concepts

Cartesian plane

The Cartesian plane is essentially a two-dimensional grid used in mathematics to plot points and lines. It consists of two axes, the x-axis (horizontal) and y-axis (vertical), which intersect at a point called the origin, denoted as (0,0).
Each point in this plane is represented by a pair of numbers, often called coordinates, written in the form (x, y). The x-coordinate tells you how far along the x-axis the point is, while the y-coordinate tells you the distance along the y-axis.
When solving problems involving iterated integrals, like the given exercise, plotting on the Cartesian plane helps to visualize the region of integration. This involves rendering lines such as y = 3 - 2x, y = x, and x = 0 to see where they intersect and form a bounded region. These intersections are crucial to setting the limits for integration.

vertical cross-sections

Vertical cross-sections involve slicing through the region vertically parallel to the y-axis. For iterated integrals, this means integrating with respect to y first and then with respect to x.
When we analyze these vertical cross-sections, we determine how y varies for each fixed x within the defined region. In the original exercise, y varies between the lines y = x and y = 3 - 2x. Therefore, for each x, the limits for y in the integral are from x to 3 - 2x.
The range for x is determined by the intersection points of these two lines with x = 0, which gives limits for x from 0 to 1. The iterated integral is therefore:

• Inner integral from x to 3 - 2x with respect to y
• Outer integral from 0 to 1 with respect to x

The integration order plays an important role in efficiently calculating the area of the bounded region.

horizontal cross-sections

Horizontal cross-sections are slices that go across the region parallel to the x-axis, meaning we integrate with respect to x first, then y.
For horizontal cross-sections in the given exercise, we observe how x varies for each fixed y within the region. x starts at 0 and goes to the line which is solved as x = (3-y)/2, considering y is between 0 and 1.
Here are the steps needed to configure the iterated integral:

• Inner integral from 0 to (3-y)/2 with respect to x
• Outer integral from 0 to 1 with respect to y

This arrangement is beneficial when the bounded region is more complex or when integrating becomes simpler after initially addressing the x-component.

bounded regions

A bounded region on the Cartesian plane is defined by its boundary lines where all points within are contained. The given problem involves three lines: y = 3 - 2x, y = x, and x = 0 (the y-axis).
These lines create an enclosed area, or bounded region, which is essential for setting limits for integration in iterated integrals.
To identify the exact bounded region, visualize where these lines intersect. Intersections are often calculated algebraically to find the range and scope for x and y values. For instance, in this case, intersections occur where y = 3 - 2x intersects y = x, giving key boundary points that guide the integration limits.
Understanding bounded regions ensures that the integral accurately measures the area, providing important context for each step of the solution process.