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Definite integrals are a key concept in calculus, helping you find the area under a curve between two specific points. Here, the points are from 0 to 1, which means we want to find the total area beneath the curve from 0 moving up to 1 along the x-axis. A definite integral can be denoted by \( \int_{a}^{b} f(x) \, dx \), where \(a\) and \(b\) are the limits of integration. This process involves evaluating the antiderivative at these points and finding their difference.

Key points of definite integrals:

• The limits, \(a\) and \(b\), provide boundaries within which the function is evaluated.
• The calculation results in a single numerical value representing the "net area" under the curve.
• The process involves using the antiderivative to simplify the evaluation.

The exercise above involved finding \( \int_{0}^{1} (x^{4/3} + x^{5/4}) \, dx \), calculating the area under the two curves within those limits.

Definite integrals are uniquely valuable because they give meaningful real-world insights, like total distance traveled by an object over time or accumulation of quantities in physics and economics.

The power rule of integration is a straightforward method for integrating polynomial expressions of the form \(x^n\). According to this rule, for any real number \(n eq -1\), the integral of \(x^n\) is given by:\[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\]Where \(C\) is the constant of integration, a necessary part of indefinite integrals.

This exercise required integrating terms like \(x^{4/3}\) and \(x^{5/4}\), applying the power rule separately to each. By increasing the exponent by 1 and dividing by the new exponent, we obtain:

• \(\int x^{4/3} \, dx = \frac{x^{7/3}}{7/3} = \frac{3}{7}x^{7/3}\)
• \(\int x^{5/4} \, dx = \frac{x^{9/4}}{9/4} = \frac{4}{9}x^{9/4}\)

The simplicity and efficiency of the power rule make it a preferred technique for tackling polynomials, a fundamental building block for more complex functions.

Evaluating integrals involves solving the integral expression to find the exact value, especially for definite integrals. Once the antiderivatives are found using integration techniques such as the power rule, we substitute the upper and lower limits of the definite integral. This involves calculating the values at the top limit and subtracting from it the value at the bottom limit.

A key step in the exercise was evaluating the definite integral from 0 to 1:

• For \(x^{4/3}\), compute \(\frac{3}{7}x^{7/3}\bigg|_{0}^{1} = \frac{3}{7}(1^{7/3} - 0^{7/3})\)
• For \(x^{5/4}\), compute \(\frac{4}{9}x^{9/4}\bigg|_{0}^{1} = \frac{4}{9}(1^{9/4} - 0^{9/4})\)
• Adding these results yields the total area: \(\frac{3}{7} + \frac{4}{9}\)

Combining the fractions requires finding a common denominator, leading to a final answer of \(\frac{55}{63}\). This closed form indicates the definite integral value over the given interval, reflecting the approach accuracy and consistency used in the evaluation.