Question

Let \(R\) be a region in the \(x y\) -plane whose area is \(B\). If \(f(x, y)=k\) for every point \((x, y)\) in \(R,\) what is the value of \(\int_{R} \int f(x, y) d A ?\) Explain.

Short answer

The value of \(\int_{R} \int f(x, y) d A\) is \(kB\).

Step-by-step solution

Understand the function

The function \(f(x, y) = k\) is constant over the region \(R\). That means, for every point \((x, y)\) in \(R\), the value of \(f\) is the same and equals to \(k\).

Calculate the double integral

Since the value of the function \(f(x, y)\) is constant and equals \(k\), and given that the area of the region \(R\) is \(B\), the value of the double integration over the region \(R\) would be \(kB\).

Result interpretation

The value of the double integral of \(f\) over the region \(R\) equals \(kB\). This means that if we add up all the values of \(f(x, y)\) for every point in region \(R\), it would sum up to \(kB\).

Key concepts

Integration

In calculus, integration is a fundamental process that can be thought of as the reverse of differentiation or as finding the total accumulation of a quantity. When you encounter an integral, you're asked to calculate the cumulative effect of a function. For instance, in the context of two-dimensional space, a double integral allows us to sum up the function's values over a certain area, giving us a total amount. This is crucial in many applications such as finding the area under a curve, calculating volumes, or even in computing work done by a force over a path.

As seen in our original problem, the concept of the double integral extends to multiple integration across a region of space to account for functions that vary over an area, instead of just along a line. When dealing with constant functions across the region, the double integral reduces to a simpler computation - multiplying the function's value by the area of the region.

Constant Function

A constant function is a type of function where the output is the same, no matter the input. In mathematical terms, if we have a two-variable function, such as our function f(x, y), and it is equal to a constant k, it means that for every x and y within the defined domain, the function's value does not change.

Visualizing this is quite simple; if you were to plot this function on a three-dimensional graph, you would get a flat plane that stands parallel to the x-y plane at a height of k. In the context of integration, constant functions are particularly straightforward to integrate over any region because the function's value remains uniform across the entire area.

Area of a Region

In geometry and calculus, the area of a region refers to the amount of two-dimensional space within a set boundary. This is a fundamental concept when working with double integrals in calculus since the integral is often used to compute the area. Whether you're dealing with simple shapes like rectangles and circles, or more complex regions bounded by curves, the double integral can be applied to find the area contained within.

Specifically, when the function being integrated is constant, as in our exercise, the area of a region simplifies the process. Since the height of the function is the same at all points (the value k), the double integral across the area B is essentially the product of the constant value k and the area B, resulting in kB, which represents the total volume under the surface f(x, y) = k above the region R.