Question

One understanding of an iterated integral is that \(\int_{a}^{b} \int_{g_{1}(x)}^{g_{2}(x)} d y d x\) gives the ______ of a plane region.

Short answer

The iterated integral gives the area of a plane region.

Step-by-step solution

Understanding the Iterated Integral

An iterated integral of the form \(\int_{a}^{b} \int_{g_{1}(x)}^{g_{2}(x)} dy \, dx\) represents a method to determine the measure of a specific quantity over a region in the xy-plane. To comprehend what this integral represents, one must recall that it sums the integrals of \(dy\) for each horizontal slice along the x-interval from \(a\) to \(b\).

Identify the Geometric Interpretation

The inner integral \(\int_{g_{1}(x)}^{g_{2}(x)} dy\) calculates the length of a vertical line (or strip) at a given \(x\), which extends from \(y = g_1(x)\) to \(y = g_2(x)\).

Calculate the Region Over x-Interval

The outer integral \(\int_{a}^{b} \, dx\) involves summing these line lengths across the interval from \(x = a\) to \(x = b\). This collective process results in the measure of the two-dimensional region enclosed by the curves \(y = g_1(x)\), \(y = g_2(x)\), and the vertical lines \(x = a\) and \(x = b\).

Conclusion on the Integral's Meaning

The iterated integral \(\int_{a}^{b} \int_{g_{1}(x)}^{g_{2}(x)} dy \, dx\) calculates the area of the region bounded by the curves \(y = g_1(x)\), \(y = g_2(x)\), and the lines \(x = a\) and \(x = b\).

Key concepts

Area Under a Curve

When discussing iterated integrals, understanding the concept of the "area under a curve" is essential. The area under a curve typically refers to the region enclosed between a graph and a reference axis, such as the x-axis on a graph. Here, it's a bit more complex because we're considering a two-dimensional area on the xy-plane rather than just a straightforward single curve along one axis.
This is handled by using double or iterated integrals. The area enclosed by curves can be found via the double integral, where we're integrating vertically within so-called "slices" of the function and then summing these slices across the desired interval horizontally.

• The first step is integrating along the variable of interest to find the area of each slice.
• Subsequently, these areas are added up over a specified range, which is handled by the outer integral.

This technique is effective for evaluating the area of more complex regions that don't necessarily follow simple geometric shapes.

Multiple Integrals

Multiple integrals are a powerful tool in calculus used to calculate things like area, volume, or even higher-dimensional quantities. Specifically, in iterated integrals, we perform integrations in a sequence - one variable at a time.
This concept is foundational in many fields, allowing the calculation of areas and volumes through successive integrations.

• The inner integral focuses on a particular variable, usually related to the height or depth of a region or function.
• The outer integral then accumulates the results of these inner integrals over another variable, often representing the breadth or length.

This process might seem intricate at first, but it's a stepwise method where each level simplifies the calculation until the overall region is accounted for. Thus, working with multiple integrals broadens the dimensions over which we calculate quantities.

XY-Plane Region

In the context of iterated integrals, the term "xy-plane region" refers to a specific area or shape in the two-dimensional plane defined by x and y axes. This region is where our calculation takes place.
These regions aren't bound to simple geometric shapes like rectangles or circles. They are more flexible, determined by the limits of the integral itself. Here, iterated integrals solve for complex shapes formed by curves such as \(y = g_1(x)\) and \(y = g_2(x)\).
To visualize it, imagine drawing lines from \(y = g_1(x)\) to \(y = g_2(x)\) for a range of \(x\) values from \(a\) to \(b\), creating an entire enclosed region in between.

• The vertical strips run from \(y = g_1(x)\) to \(y = g_2(x)\)
• The area between these strips and over the entire horizontal interval from \(x = a\) to \(x = b\) is then calculated

This conception of analyzing and calculating predefined regions on the xy-plane underpins many practical applications of calculus.

Geometric Interpretation of Integrals

The geometric interpretation of integrals builds an intuitive understanding of what an iterated integral is essentially doing. Each individual component of the integration process can be visualized geometrically.
Take the inner integral \(\int_{g_1(x)}^{g_2(x)} dy\), which represents the height of a vertical line at each x-value within the region being evaluated. This shows one "slice" or "strip" of the region. The outer integral \(\int_{a}^{b} dx\) then calculates the full area by summing the heights of these slices from \(x = a\) to \(x = b\).

• Each strip gives you a height from the curves to the x-axis.
• These heights, when summed over the range, give a total area.

In essence, this two-step integration visually represents how areas form by layering numerous narrow strips. This interpretation not only aids in simple calculations but also helps in grasping more complex multi-dimensional calculus operations.