Problem 28 - Thomas Calculus

Question

If \(y = f ( x )\) is a nonnegative continuous function over the closed interval \(a \leq x \leq b ,\) show that the double integral definition of area for the closed plane region bounded by the graph of \(f ,\) the vertical lines \(x = a\) and \(x = b ,\) and the \(x\) -axis agrees with the definition for area beneath the curve in Section \(5.3 .\)

Step-by-Step Solution

Answer

The area found using the double integral \( \int_{a}^{b} f(x) \ dx \) matches the definition of area under the curve from Section 5.3.

Step 1: Understanding the Double Integral Definition of Area

The double integral for the area under a curve over a region is given by \( \iint_{R} 1 \ dA \), where \( R \) is the region beneath the curve of \( y = f(x) \), between \(x = a\) and \(x = b\). Thus, this integral equals \( \int_{a}^{b} \int_{0}^{f(x)} 1 \ dy \ dx \). This involves integrating with respect to \( y \), then \( x \).

Step 2: Evaluate the Inner Integral

The inner integral \( \int_{0}^{f(x)} 1 \ dy \) results in \( f(x) \), because the integral of \(1\) with respect to \(y\) from \(0\) to \(f(x)\) is the length of the interval \([0, f(x)]\).

Step 3: Evaluate the Outer Integral

Now, the outer integral becomes \( \int_{a}^{b} f(x) \ dx \). Integrating \(f(x)\) with respect to \(x\) over \([a, b]\) gives the area under the curve as defined in Section 5.3.

Step 4: Conclusion

Both the double integral and the single variable integral definitions yield the same area \( \int_{a}^{b} f(x) \ dx \), confirming the equivalence of the two definitions for the area under a curve.

Key Concepts

Double Integral

A double integral is a fundamental concept in calculus, especially when dealing with functions of two or more variables. It extends the idea of an integral to higher dimensions. In the original exercise, we use a double integral to find the area beneath a function. This involves two integration processes.

• First, it's performed over the variable that corresponds to the height of the area (usually the y-axis).
• Then, it's done over the base of that area, typically the x-axis, to cover the region fully.

This process is represented by the symbol \( \iint \). The double integral \( \int_{a}^{b} \int_{0}^{f(x)} 1 \ dy \ dx \) helps us visualize how each small portion of area is first considered (taken over y) and then summarized (over x). Double integrals are crucial for calculating

• areas of complex surfaces,
• volumes under surfaces, and
• averages over regions.

Area Under a Curve

Finding the area under a curve is a classical problem in calculus. It helps in understanding quantities like total distance or total accumulated change. The area under the curve represents the integral of a function over a specified interval.

• In a single integral, it calculates the area beneath the function graph directly, as seen in the formula \( \int_{a}^{b} f(x) \ dx \).
• In the context of the double integral from our problem, it first calculates the area vertically (inner integral along y) and then horizontally (outer integral along x).

The equivalence between these two methods is demonstrated by showing that both approaches give the same result.

Continuous Function

A continuous function is fundamental in calculus because it guarantees the function behaves predictably over an interval. Continuity in mathematical terms means

• there are no breaks, jumps, or holes in the graph of the function,
• and the function can be drawn without lifting the pen off the paper.

For the area under a curve calculation, continuity ensures:

• the integral will have a well-defined value over the interval \([a, b]\),
• and that definitions using single or double integrals remain consistent.

A non-negative continuous function over a closed interval is required in the problem to properly apply both the double integral and the single integral definitions. Without continuity, the area computation might be invalid or undefined during certain points.

Integrals

Integrals are a core part of calculus, representing the accumulation of quantities and the notion of adding up infinitesimally small pieces. In this context, integrating a function involves:

• breaking down the problem into small parts (infinitesimals), and
• adding them together to understand the whole.

Specifically, the single integral \( \int_{a}^{b} f(x) \ dx \) finds the total area under a curve, while the double integral \( \int_{a}^{b} \int_{0}^{f(x)} 1 \, dy \, dx \) shows a more detailed process of building that area.In calculus, integrals:

• solve problems involving rates of change by finding total accumulated values,
• are used in physics for concepts like work and energy, and
• provide the necessary tools for analyzing curves and surfaces in more complex, multi-dimensional spaces.

Thomas Calculus

Thomas Calculus is a popular textbook among students learning calculus. It provides a thorough introduction to the foundational concepts, including integrals and their applications. In this context:

• Thomas Calculus helps explain the equivalence of different definitions for the area under a curve, as shown in sections like 5.3.
• It emphasizes visual understanding through graphs and intuitive explanations.
• By using established examples and exercises, students can connect the theoretical with the practical, seeing how integrals operate in real-world problems.

The use of a standardized resource like Thomas Calculus ensures that learners have a reliable framework to study from, which is particularly beneficial when dealing with multi-step processes like understanding double integrals.