Question

Write an iterated integral of a continuous function \(f\) over the following regions. The region bounded by the triangle with vertices \((0,0),(2,0),\) and (1,1)

Short answer

Based on the step-by-step solution above, write an iterated integral for a continuous function f over the triangular region with vertices (0,0), (2,0), and (1,1). Answer: \(\int_{0}^{1} \int_{0}^{x} f(x,y) \mathrm{dy} \mathrm{dx} + \int_{1}^{2} \int_{0}^{-x+2} f(x,y) \mathrm{dy} \mathrm{dx}\)

Step-by-step solution

Determine the equations of the sides of the triangle

First, let's determine the equations for the three lines that form the triangle: 1. Line between (0,0) and (2,0): This line is along the x-axis, so y = 0 2. Line between (0,0) and (1,1): The slope of this line is 1/1 = 1, so the equation is y = x 3. Line between (2,0) and (1,1): The slope of this line is 1/-1 = -1, so the equation is y = -x + 2 Now we have the equations of all three sides: - y = 0 (x-axis) - y = x - y = -x + 2

Determine the limits of integration for x and y

Now let's determine the limits of integration for x and y by considering the region bounded by the three lines. The triangle lies above the x-axis (y = 0) and is bounded by the lines y = x and y = -x + 2. To find the intersection between these two lines, we can set them equal to each other: x = -x + 2 Solve for x: 2x = 2 x = 1 Now we know that the region lies between x = 0 and x = 2 since all the vertices have x-coordinates between these values. Additionally, we can consider the bounded region for each value of x separately: For x between 0 and 1, the region will be bounded by y = 0 and y = x. For x between 1 and 2, the region will be bounded by y = 0 and y = -x + 2.

Set up the iterated integral

Now that we have determined the limits of integration for both x and y, we can set up the iterated integral. We will write the integral with respect to y first, then x: \(\int_{0}^{1} \int_{0}^{x} f(x,y) \mathrm{dy} \mathrm{dx} + \int_{1}^{2} \int_{0}^{-x+2} f(x,y) \mathrm{dy} \mathrm{dx}\) This iterated integral represents the continuous function f over the given triangular region.

Key concepts

Calculus

Calculus is a branch of mathematics that studies how things change. It provides tools to model and understand dynamic systems, such as the rate at which a car accelerates or how market prices fluctuate over time. In calculus, we often deal with quantities that continuously vary. The fundamental concepts of calculus include:

• Limits: These are used to define derivatives and integrals. They help us understand how functions behave as their inputs approach certain values.
• Derivatives: These measure how a function changes as its input changes. It's like finding the slope of a curve at any point.
• Integrals: These are used to find areas under curves, among other things. They are the reverse process of taking a derivative.

In our iterated integral exercise, calculus allows us to setup and evaluate the area under a surface defined by a function over a specific region. This region is a triangle described by certain lines in the coordinate plane.

Limits of Integration

The limits of integration tell us the boundaries over which we're performing our integral calculations. For an iterated integral, we first set boundaries for the inner integral, and then for the outer integral.

In this example, we are working over a triangular region with vertices at \((0,0), (2,0), (1,1)\), resulting in various limits for the integrals:

• Inner Integral: This is performed with respect to \(y\), the vertical bounds of the triangle region. For \(x\) between 0 and 1, \(y\) ranges from 0 to \(x\). For \(x\) between 1 and 2, \(y\) ranges from 0 to \(-x+2\).
• Outer Integral: This is performed with respect to \(x\), the horizontal bounds, which range from 0 to 2, covering the entire base of the triangle.

These specific bounds ensure that the entire triangular region is covered by the iterated integral.

Continuous Function

A continuous function is one that has no breaks, jumps, or holes in its graph. This means that you can draw the function's graph without lifting your pencil off the paper.

In calculus, continuous functions are important because they ensure we can perform operations like differentiation and integration smoothly. They provide predictability and reliability, allowing the geometry underlying the function to be fully explored.

• Definition: Mathematically, a function \(f(x)\) is continuous at a point \(c\) if \(\lim_{x \to c} f(x) = f(c)\). This means as we get infinitely close to \(c\), the function approaches \(f(c)\).
• Role in Integrals: When setting up iterated integrals, the continuity of a function guarantees that the surface area within the bounds can be smoothly calculated.

For our iterated integral over the triangular region, assuming \(f(x, y)\) is continuous, ensures that the calculations of the area or other values below the surface are valid and complete.