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Definite integrals are a type of integral that calculates the area under a curve within a specific interval. Unlike indefinite integrals, which provide a general antiderivative, definite integrals have set upper and lower limits. These limits are often referred to as the boundaries of integration. When evaluating a definite integral, the result is a number that represents the accumulated quantity, such as area or total change, between the specified limits.

The notation for definite integrals uses a function and its limits of integration, like so:

• The expression \( \int_{a}^{b} f(x) \, dx \) where \( a \) and \( b \) are the lower and upper limits, respectively.
• This integral computes the total area under the curve \( f(x) \) from \( x = a \) to \( x = b \).

To solve a definite integral, you usually find an antiderivative first and then apply the Fundamental Theorem of Calculus. The theorem states that if \( F \) is an antiderivative of \( f \), then the integral can be evaluated as \( F(b) - F(a) \), where \( F(b) \) and \( F(a) \) are the values of the antiderivative at the bounds.

Trigonometric functions are functions related to angles commonly found in mathematics, particularly when dealing with triangles and waves. Common trigonometric functions include sine, cosine, and tangent.

In the context of integrals,

• Sine and cosine functions appear frequently, often as part of periodic or oscillating systems.
• Their integrals can be used to find amplitudes and transformations of waves.

For instance, in our exercise, the integral \( \int \sin^2(x) \, dx \) emerges from the need to calculate areas involving a sinusoidal wave.

A key property of trigonometric functions, especially sine and cosine, is their periodic nature. Over a complete cycle, such as from 0 to \(2\pi\) for sine and cosine, these functions repeat their values. When finding definite integrals involving trigonometric functions, understanding their periodicity can simplify the calculation by showing symmetries or repetitions.

Limits of integration are the bounds between which we evaluate a definite integral. They determine the interval over which we calculate the area under a curve. Changing these limits changes the result of the integral, as we look over a different section of the function.

In substitution, the limits also undergo transformation based on the substituted variable. For example, in the original exercise, we perform a substitution \( u = 3x \), which affects the limits:

• When \( x = 0 \), \( u = 0 \).
• When \( x = \frac{\pi}{3} \), \( u = \pi \).

Thus, the integral \( \int_{0}^{\frac{\pi}{3}} 3 \sin^2(3x) \ dx \) transfers to \( \int_{0}^{\pi} \sin^2(u) \ du \).

This method shows how, through substitution, limits help maintain the equality of integrals by adjusting the variable and the expressed interval, ensuring that the resulting integrals cover equivalent areas despite the transformation. Understanding limits of integration is essential for correctly setting up and evaluating definite integrals.