Problem 49 - Thomas Calculus

Question

Suppose \(f(x)\) is positive, continuous, and increasing over the interval \([a, b] .\) By interpreting the graph of \(f\) show that $$ \int_{a}^{b} f(x) d x+\int_{f(a)}^{f(b)} f^{-1}(y) d y=b f(b)-a f(a) $$

Step-by-Step Solution

Answer

The integrals cover the rectangle area minus overlap, proving the equation.

Step 1: Understand the Expression

We have the expression to prove, which combines two integrals and an equation: \( \int_{a}^{b} f(x) \, dx + \int_{f(a)}^{f(b)} f^{-1}(y) \, dy = b f(b) - a f(a) \). The task is to interpret this expression graphically.

Step 2: Analyze the Integral of \(f(x)\)

The integral \( \int_{a}^{b} f(x) \, dx\) represents the area under the curve of \( f(x) \) from \( x=a \) to \( x=b \). This area is under the curve on the \( x-y \) plane.

Step 3: Analyze the Integral of \(f^{-1}(y)\)

The integral \( \int_{f(a)}^{f(b)} f^{-1}(y) \, dy \) represents the area under the curve of \( f^{-1}(y) \) from \( y=f(a) \) to \( y=f(b) \). This area is in the inverse curve on the \( y-x \) plane.

Step 4: Understanding the Rectangle Area

Consider the rectangle with corners at \((a, f(a)), (a, f(b)), (b, f(a)), (b, f(b))\). The area of this rectangle is \( (b-a) \times (f(b) - f(a)) = b f(b) - a f(a) + a f(b) - b f(a) \).

Step 5: Relate Areas to Prove Equality

The sum of the areas represented by \( \int_{a}^{b} f(x) \, dx \) and \( \int_{f(a)}^{f(b)} f^{-1}(y) \, dy \) together cover all but the overlap of the rectangle. The overlapping area is added twice, which is \( a f(b) - b f(a) \). Hence, the sum of the integrals equals the total rectangle area minus the overlap once, resulting in the equation \( b f(b) - a f(a) \).

Key Concepts

Area Under a Curve

In integral calculus, the term "area under a curve" is essential when understanding integrals. Consider a continuous and increasing function, like \( f(x) \), over a specific interval \([a, b]\). The integral \( \int_{a}^{b} f(x) \, dx \) represents the area between the curve of \( f(x) \) and the x-axis from \( x = a \) to \( x = b \). This area is visually the region beneath the curve and above the x-axis.

• This concept is pivotal because it allows us to quantify the accumulated value or total area under the function on this interval.
• Practically, this is useful in scenarios like calculating distance traveled over time or evaluating total sales revenue over a period.

Understanding the area under the curve for different functions helps in solving not only theoretical problems but also practical real-world problems.

Inverse Functions

Inverse functions are fascinating in their way of flipping inputs and outputs. For a function \( f(x) \) that is strictly increasing, its inverse \( f^{-1}(y) \) reflects the interchange of the role of \( x \) and \( y \). This means that if \( f(a) = y \), then \( f^{-1}(y) = a \).
The inverse function \( f^{-1}(y) \) graphs in the \( y-x \) plane, effectively swapping the x and y axes.

• Because \( f \) and \( f^{-1} \) are related through this interchange, understanding one aids in understanding the other.
• Integral of \( f^{-1}(y) \), \( \int_{f(a)}^{f(b)} f^{-1}(y) \, dy \), mirrors the area translation under this inverse curve on the \( y-x \) plane from \( y = f(a) \) to \( y = f(b) \).

This insight is not only mathematically intriguing but also practical when reversing effects or predictions in data modeling.

Definite Integrals

Definite integrals are crucial in integral calculus as they provide the total accumulated change across an interval. For example, \( \int_{a}^{b} f(x) \, dx \) computes the exact area under the curve of \( f(x) \) from \( x = a \) to \( x = b \).
In our case, we also consider \( \int_{f(a)}^{f(b)} f^{-1}(y) \, dy \), doing a similar job but for the inverse function.

• Definite integrals require clear boundaries or limits, as in the upper and lower bounds of the integral.
• They provide concrete values rather than functions, helping solve equations involving areas and accumulated quantities.

Thus, definite integrals bridge the gap between pure mathematics and its application in finding total quantities.

Graphical Interpretation

Graphical interpretation is a powerful way to visualize and understand mathematical concepts like integrals and areas. In this scenario, we consider \( f(x) \) and its inverse \( f^{-1}(y) \) over given intervals, plotting them on separate planes.
Graphically, integral calculus allows us to visualize these as areas:

• The area under \( f(x) \) on the x-y plane correlates with accumulated value from \( a \) to \( b \).
• The area under \( f^{-1}(y) \) on the y-x plane explains inverse behavior and integrates from \( f(a) \) to \( f(b) \).

Linking these graphically helps see how different integrals relate to the rectangle defined by the function endpoints and effectively communicate mathematical equations visually. This interpretation is not only useful for solving problems but also in enhancing comprehension through visual aids.