Question

Evaluate the definite integral. $$ \int_{0}^{1} \frac{x-\sqrt{x}}{3} d x $$

Short answer

The result of our definite integral, \(\int_{0}^{1} \frac{x-\sqrt{x}}{3} dx\), is \(-\frac{1}{18}\)

Step-by-step solution

Break Down The Integral

First, simplify the integral by expressing it as a sum of separate integrates: \[\int_{0}^{1} \frac{x-\sqrt{x}}{3} dx = \frac{1}{3} \int_{0}^{1}x dx - \frac{1}{3} \int_{0}^{1}\sqrt{x} dx\]

Computing the Integrals

Calculate the definite integrals separately. \[\frac{1}{3} \int_{0}^{1}x dx = \frac{1}{3} \cdot [\frac{x^{2}}{2}]_{0}^1 = \frac{1}{2} \cdot \frac{1}{3}\] and \[\frac{1}{3} \int_{0}^{1}\sqrt{x} dx = \frac{1}{3} \cdot [\frac{2}{3} x^{3/2}]_{0}^1 = \frac{2}{9}\]

Subtract The Integrals

Subtract the result of the second integral from the first one, as mandated by our breakdown in Step 1: \[ \frac{1}{6} - \frac{2}{9} = -\frac{1}{18}\]

Key concepts

Integral Computation

Understanding how to compute a definite integral is a fundamental skill in calculus. A definite integral represents the area under the curve of a function within specified bounds, in this case from 0 to 1. The process involves finding the antiderivative (also known as the indefinite integral) of the given function and then evaluating this antiderivative at the upper and lower limits of integration. It's like finding the total sum of infinite tiny slices under the function's curve between those two points.

For the integral given, \[ \int_{0}^{1} \frac{x-\sqrt{x}}{3} d x \], we break it down into simpler parts to make computation easier. By calculating each piece individually and then combining the results, we reach the final value of the whole area under the curve. Remember, patience and careful calculations are key to avoiding mistakes in this process.

Algebraic Expression Simplification

Simplification of algebraic expressions is a technique used to make equations easier to work with. In integral computation, it often involves breaking down complex expressions into simpler ones that are more straightforward to integrate.

For instance, in our exercise, the fraction inside the integral \[ \frac{x-\sqrt{x}}{3} \] is simplified by separating it into two different integrals. This is possible because integration, like differentiation, is linear, meaning that the integral of a sum is the sum of the integrals. Simplifying expressions before integrating prevents potential miscalculations and makes further algebraic steps more manageable.

Sqrt Function Integration

Integrating square root functions, often noted as \(\sqrt{x}\), can sometimes be challenging. To handle such integrations correctly, it is useful to remember that \(\sqrt{x}\) is the same as \(x^{1/2}\).

In this exercise, we integrated \(\sqrt{x}\) by converting it to \(x^{1/2}\) and applying the power rule of integration, which states that the integral of \(x^n\) with respect to \(x\) is \(\frac{x^{n+1}}{n+1}\), provided that \(neq -1\). After the application of this rule, we arrive at \[\frac{2\sqrt{x^3}}{3}\] for the antiderivative. Always rewriting the square root in exponential form is a straightforward technique that consistently simplifies the integration process.

Definite Integral Properties

Definite integrals have fundamental properties that make solving them more systematic. One such property is the linearity of integration, which allows us to split the integral of a sum into the sum of integrals. Another is the evaluation step, which involves finding the difference between the antiderivative's values at the upper and lower limits.

In our calculation, we used these concepts to find the area under \(\frac{x-\sqrt{x}}{3}\) from 0 to 1. We simplified the function into more basic components, integrated each part, and then combined the results. Knowing and utilizing definite integral properties are crucial for accurate and efficient solutions to integral problems.