Question

Let \(\mathcal{C}\) denote the set of points that are interior to, or on the boundary of, a square with opposite vertices at the points \((0,0)\) and \((1,1)\). Let \(Q(C)=\iint_{C} d y d x\). (a) If \(C \subset \mathcal{C}\) is the set \(\\{(x, y): 0

Short answer

(a) \(Q(C) = 1/2\) (b) \(Q(C) = 0\) (c) \(Q(C) = 1/3\).

Step-by-step solution

Steps for (a)

(a) Here, \(C\) is defined as \(\{(x, y): 0<x<y<1\}\). In other words, this represents a right triangle defined by the x-axis, the line \(y=x\) and line \(y=1\). To find the double integral, we iterate it as two single integrals. We first integrate with respect to \(y\) where the lower limit is \(x\) and upper limit is \(1\). Then integrate with respect to \(x\) where the limits are \(0\) and \(1\). So, \(Q(C)=\int_{0}^{1} \int_{x}^{1} dy dx\). By carrying out these integrations, we get \(Q(C) = 1/2\).

Steps for (b)

(b) Here, \(C\) is defined as \(\{(x, y): 0<x=y<1\}\), i.e., it is the diagonal line of the square. However, since we are calculating a double integral, and the line segment is of zero area, \(Q(C) = 0\).

Steps for (c)

(c) Here, \(C\) is defined as \(\{(x, y): 0<x / 2 \leq y \leq 3 x/ 2<1\}\). This represents a strip between the lines \(y = x/2\) and \(y = 3x/2\). For the bounds of \(x\), note that the strip touches the square at the points \((0,0)\), \((2/3,1)\), and \((1,1/2)\), which gives us \((0\leq x \leq 2/3)\) for the region below \(y = 1\) and \((2/3 \leq x \leq 1)\) for the region above \(y = 1\). So, we calculate the double integral as a sum of two integrals, \(Q(C) = \int_0^{2/3} \int_{x/2}^{3x/2} dy dx + \int_{2/3}^{1} \int_{x/2}^{1} dy dx\). This gives us \(Q(C) = 5/18 + 1/18 = 1/3\).

Key concepts

Mathematical Statistics

Mathematical statistics is a key subset of statistics where the focus lies on developing new statistical methods using mathematics. It involves the study of probability theory to interpret datasets and make inferences about populations from samples.

For example, in computing the measurement of a region in probability or in sampling, the double integral is an essential tool. It allows us to sum up values over a two-dimensional area, giving us insights into probabilities, mean values, and other important statistical measures over a region such as set \(C\) from the exercise.

The double integral can be particularly powerful when the set \(C\) is described by inequalities, such as \(0
In our exercise, we've applied double integrals to calculate \(Q(C)\), which in a real-world scenario could correspond to the likelihood of a random point falling within the region \(C\). Furthermore, double integrals are used to compute the expected value of a function over a two-dimensional space which is a fundamental concept in mathematical statistics.

Integration

Integration in mathematics is a fundamental concept used to find areas, volumes, central points, and many other useful things. The double integral, specifically, extends the idea of integration to calculate the volume under the surface over a region in the xy-plane.

In our exercise, the double integral \(\iint_{C} dy dx\) is used to calculate the 'area' occupied by set \(C\) within the square. When set \(C\) is the triangle described by \(0
It's important to note that the zero area result for part (b) reflects a property of integration: when integrating over a line or curve with respect to two dimensions, the result is zero, since a line does not have an 'area' in two dimensions. Thus, understanding how to set up and evaluate double integrals is critical for solving problems in various fields of mathematics and the sciences.

Interior Points

The concept of interior points is fundamental in understanding the geometric regions over which we often integrate. An interior point is any point within a given set that has a neighborhood completely contained within the set. Understanding the interior points of a set helps establish the limits of integration.

In the context of our exercise, interior points of set \(\mathcal{C}\), a square, are critical in identifying which points lie within the boundaries necessary for the computation of \(Q(C)\). For instance, the triangle in part (a) consists only of interior points of the square. Identifying the interior points of \(C\) in the inequalities provided guides us in setting the correct bounds for our double integral.

For part (c), it becomes clear upon visual inspection which points of the strip between \(y = x/2\) and \(y = 3x/2\) are actually interior points of the square \(\mathcal{C}\). Thus, determining the limits for \(x\) when \(y\) equals 1 is crucial. By pinpointing where the strip intersects the square's boundary, we can correctly establish the piecewise bounds for the double integral necessary to calculate \(Q(C)\). Understanding interior points is not only useful for integration but is also a foundation for topology, a branch of mathematics dealing with properties of space that are preserved under continuous transformations.