Suppose f(x) is positive, continuous, and increasing over...

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

Key Concepts

Inverse Functions

An inverse function essentially reverses the role of inputs and outputs of the original function. When a function is strictly increasing and continuous, its inverse exists and is also continuous and strictly increasing. Graphically, the curve of an inverse function is the reflection of the original function’s curve across the line y = x. This concept allows one to relate areas under the curves of the function and its inverse.

Definite Integrals as Areas

A definite integral represents the area under the curve of a function over a specified interval. By interpreting integration geometrically, one can visualize the accumulation of area between the function's graph and the x-axis. This interpretation is pivotal in understanding how areas corresponding to a function and its inverse combine to yield a geometrical relationship involving rectangular regions.

Geometric Interpretation of Inverse Relationship

By analyzing the graph of a function and its inverse, one can see that the sum of the areas under each curve over corresponding intervals equals the area of a rectangle determined by the end-values of the function. This relationship is encapsulated in the identity involving the integrals of f and f?¹, illustrating how the complementary areas combine to form a full rectangle with one side corresponding to the x-values and the other to the y-values.

Change of Variables in Integration

Changing the variable of integration is an essential technique that facilitates the evaluation of integrals, especially when dealing with inverse functions. Through a variable substitution, one can switch between the function and its inverse, aligning the limits of integration accordingly. This process underpins the derivation of the formula that equates the sum of the integrals of a function and its inverse to the area of a corresponding rectangle.

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