Prove, in the following two ways, that for any integer k,...

Prove, in the following two ways, that for any integer $k$, the signed area under the graph of the function $f(x)=$ $\sin (2(x-(\pi / 4)))$ on the interval $[0, k \pi]$ is always zero: (a) by calculating a definite integral; (b) by considering the period and graph of the function $f(x)=\sin (2(x-(\pi / 4)))$.

Prove, in the following two ways, that for any integer $k$, the signed area under the graph of the function $f(x)=$ $\sin (2(x-(\pi / 4)))$ on the interval $[0, k \pi]$ is always zero:
(a) by calculating a definite integral;
(b) by considering the period and graph of the function $f(x)=\sin (2(x-(\pi / 4)))$.

Key Concepts

Substitution in Integration

The substitution method in integration is a technique used to simplify the integral by changing variables. In this problem, substituting for the transformed argument of the sine function converts the integral into a standard form, making it easier to calculate and showing that the integrated areas cancel over complete periods.

Symmetry and Cancellation

Symmetry in functions, particularly in sine functions which are odd, implies that the areas above and below the x-axis are mirror images over symmetric intervals. When these areas are summed over a full period or integer multiples of periods, the positive and negative contributions cancel out, yielding a total area of zero.

Definite Integration

Definite integration is a method used to compute the signed area under the curve of a function between two limits. In the context of this problem, evaluating the definite integral of the sine function over an interval that is a multiple of a certain length allows us to see that the positive and negative contributions cancel each other exactly, resulting in a net area of zero.

Periodic Functions

Periodic functions repeat their values at regular intervals, called periods. In this problem, the sine function, after appropriate transformation (a phase shift and scaling), still maintains this periodic behavior. When integrated over intervals that cover one or more complete periods, the areas above and below the x-axis cancel out, leading to a net area of zero.

Graph Transformations

Graph transformations such as horizontal shifts and stretching or compressing the input affect the position and period of a function. Specifically, shifting the input of the sine function changes its phase while scaling the input alters its period. However, these changes do not disturb the symmetry of the sine function over complete periods, which is key to understanding why the net signed area is zero.

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