91Ó°ÊÓ

Integration is a fundamental concept in calculus, and the power rule of integration is one of the simplest and most frequently used methods for finding the integral of polynomial functions. When you have a function of the form \(x^n\), the power rule states that to integrate this function, you increase the exponent by one and then divide by the new exponent. In mathematical terms, the integral can be written as: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \] where \(C\) is the constant of integration, which is typically omitted in definite integrals as it cancels out. In our exercise, for each term, \(x^{1/2}\) and \(x^{1/4}\), we apply the power rule by finding their integrals:

• For \(x^{1/2}\): Increase the power to \(3/2\), resulting in \(\frac{2}{3}x^{3/2}\).
• For \(x^{1/4}\): Increase the power to \(5/4\), resulting in \(\frac{4}{5}x^{5/4}\).

This is the crucial step towards evaluating definite integrals of polynomial terms.

Definite integrals differ from indefinite integrals primarily as they evaluate the accumulation of quantities, often represented as the area under a curve between two limits. When working with a definite integral, it's important to specify the upper and lower bounds of integration, noted as \(a\) and \(b\). The integral is calculated from \(a\) to \(b\), producing a concrete numerical result. For our problem, the definite integral is \(\int_{0}^{4}\), meaning we will find the values of the integrated functions at these bounds and subtract the lower bound value from the upper bound value. After applying the power rule, we substitute the boundaries into our results:

• Evaluate \(\frac{2}{3}x^{3/2}\) at \(x=4\) and \(x=0\).
• Evaluate \(\frac{4}{5}x^{5/4}\) at \(x=4\) and \(x=0\).

Calculating these will give us the total area under the curve \(x^{1/2} + x^{1/4}\) from 0 to 4.

In calculus, the sum of functions refers to the addition of two or more functions to create a new function. This technique is often used when dealing with multiple terms within an integral or derivative. The integral of the sum of functions property simplifies the computation into manageable parts by allowing us to integrate each function individually. In mathematical terms, if you have two functions \(f(x)\) and \(g(x)\), then: \[ \int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx \] Our exercise consists of the sum of two functions: \(x^{1/2}\) and \(x^{1/4}\). By applying this property, we break the integral \(\int_{0}^{4}(x^{1/2} + x^{1/4}) \, dx\) into separate integrals:

• \(\int_{0}^{4} x^{1/2} \, dx\)
• \(\int_{0}^{4} x^{1/4} \, dx\)

This step is vital as it allows for individual application of integration rules, like the power rule, to each component without interference from the other terms.