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Chapter 14: Problem 14

Sketch the following regions and write an iterated integral of a continuous function \(f\) over the region. Use the order \(d y d x\) \(R\) is the triangular region with vertices \((0,0),(0,2),\) and (1,1)

Short Answer

Expert verified
Question: Write an iterated integral for a continuous function f over the triangular region R defined by the vertices (0,0), (0,2), and (1,1) using the order dy dx. Answer: The iterated integral for a continuous function f over the triangular region R is given by: $$\int_{0}^{1}\int_{0}^{-x+2}f(x,y)\,dy\,dx$$

Step by step solution

01

Sketch the triangular region R

First, plot the given vertices on a Cartesian plane: - (0,0) is the origin - (0,2) is the point on the y-axis where y = 2 - (1,1) is the point where x = 1 and y = 1 Next, draw lines connecting these vertices to form a triangle. You should have a triangle with vertices at (0,0), (0,2), and (1,1).
02

Determine the equations of the sides

Now we need to find the equations of the lines forming the sides of the triangle: 1. For the vertical line segment connecting (0,0) and (0,2), the equation is x = 0. 2. For the diagonal line segment connecting (0,2) and (1,1), we can use the two-point form to find the line equation: (y-2) / (x-0) = (1-2) / (1-0) => y = -x + 2. 3. For the diagonal line segment connecting (1,1) and (0,0), we can use the two-point form to find the line equation: (y-0) / (x-1) = (1-0) / (1-0) => y = x.
03

Write the limits of integration

We have our region R formed by the triangle, and we want to set up an iterated integral with the order dy dx. 1. The lower limit for x is the left boundary of the region, which is the vertical line x = 0. 2. The upper limit for x is the right boundary of the region, which is the diagonal line y = x. 3. The lower limit for y is the bottom boundary of the region, which starts from y = 0 and goes to the diagonal line y = -x + 2. So the y-bounds for the integral are from 0 to -x + 2.
04

Write the iterated integral

Now we have all the information we need to write the iterated integral of a continuous function f over the region R, in the order dy dx: $$\int_{0}^{1}\int_{0}^{-x+2}f(x,y)\,dy\,dx$$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Triangular Region
When working with iterated integrals, identifying the region of integration is crucial. In this exercise, the region of interest is a triangular one.
A triangular region is a shape on the plane with three straight sides. Here, our triangle is formed by the vertices (0,0), (0,2), and (1,1).
Each vertex represents a corner of the triangle, and the sides are the straight lines connecting these points. To better understand:
  • The line between (0,0) and (0,2) is a vertical line tracing along the y-axis.
  • The diagonal line between (0,2) and (1,1) slopes downward, and its equation can be determined using a straight line formula.
  • The other diagonal line between (0,0) and (1,1) has an upward slope.
Visualizing these connections can help sketch the triangular region accurately on a Cartesian plane. This step forms the basis for setting up correct integration limits later.
Limits of Integration
Defining the limits of integration is essential for solving iterated integrals over a given region. These limits define the scope of x and y values for the function you intend to integrate.
In this context, we use the order of integration \(dy \, dx\), meaning we integrate with respect to \(y\) first, and then \(x\).
The steps to determine the limits for this triangular region are:
  • The x-boundaries are determined by the vertical edges of the triangle. They start at \(x = 0\) and end at \(x = 1\), where the vertices influence these boundaries.
  • The y-limits, varying for each x, start from the lower triangle edge to the slanted upper boundary. For each specific x, \(y\) varies from 0 to the line \(y = -x + 2\).
This methodology ensures you are capturing the entire triangular region in your integration and specifies where the function \(f(x, y)\) needs to be evaluated.
Continuous Function
Understanding the concept of a continuous function is vital when setting up an iterated integral. A function is considered continuous if you can draw it without lifting your pencil, meaning there are no breaks, jumps, or disjointed sections.
For our iterated integral \(\int_{0}^{1}\int_{0}^{-x+2}f(x,y)\,dy\,dx\), the function \(f(x, y)\) needs to be continuous over the triangular region defined by the vertices.
The importance of continuity lies in ensuring the integral is computable and that it represents the area under the curve accurately within the triangular domain.
  • Continuous functions maintain all values within the region, ensuring a smooth curve.
  • Discontinuities might cause the integral to diverge or produce inaccurate results.
Thus, confirming that \(f(x, y)\) is continuous over the region solidifies the foundation for applying the iterated integral confidently.

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