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The substitution method is a fundamental technique in integral calculus used when dealing with complicated integrals. This method involves substituting part of the integral with a new variable to simplify the calculation. It is particularly useful when you have a composite function where the inner function's derivative resembles part of the integrand. Here's how it typically works:

• Identify a substitution: You will look for a part of the integral that can be represented by a new variable, usually denoted as \( u \). This substitution should help transform the integral into a simpler form.
• Compute the derivative: Once you've decided on \( u \), find the derivative \( du \) in terms of the original variable, often \( dr \), which will allow you to substitute all parts of the integral.
• Substitute and integrate: Replace the identified parts of your integral with \( u \) and \( du \) to simplify the expression, then proceed with integration.

In this exercise, the substitution method simplified the given integral by letting \( u = 4 + r^2 \), resulting in \( du = 2r \, dr \), making it more straightforward to solve.

Definite integrals involve computing the area under a curve within specified limits, providing a numerical value of the total accumulation over an interval. This area is bounded by the graph of the function, the x-axis, and vertical lines at the specified limits of integration. These limits are crucial because they determine where the evaluation starts and ends.To compute a definite integral using the substitution method:

• Convert limits: After substituting, the original limits of integration also need to be translated to the new variable. For example, when \( r = 0 \), the limit becomes \( u = 4 \); when \( r = 1 \), \( u = 5 \).
• Evaluate: After integration, substitute the upper and lower limits back into the antiderivative to find the definite integral's value. The result is a specific number that represents the net area.

In our case study, the limits for \( u \) ranged from 4 to 5, leading to a final calculated value of \( \frac{1}{8} \).

Calculus theorems like the Fundamental Theorem of Calculus connect differentiation and integration, offering a deep understanding of how accumulation and rates of change relate. It asserts that if a function is continuous over a given interval, then its integral can be evaluated by finding its antiderivative.Within this context, the theorem simplifies the process of evaluating definite integrals by linking it to finding antiderivatives:

• Compute an antiderivative: The process of integration results in an antiderivative, a function whose derivative matches the original integrand.
• Relate to definite integrals: Use the limits of integration to evaluate the difference in values of the antiderivative at these points. This result corresponds to the definite integral's value.

This exercise leverages the power of these theorems, integrating the simplified form \( \int_{4}^{5} \frac{1}{u^2} \), using its antiderivative \( -\frac{1}{u} \) to calculate the area between \( u = 4 \) and \( u = 5 \), resulting in \( \frac{1}{8} \). This showcases how these foundational concepts work together to solve complex integrals.