Problem 7
Question
Evaluate the integrals $$ \int_{0}^{1}\left(x^{2}+\sqrt{x}\right) d x $$
Step-by-Step Solution
Verified Answer
The integral evaluates to 1.
1Step 1: Break Down the Integral
The given integral can be split into two separate integrals for easier evaluation. Therefore, we have: \[\int_{0}^{1}\left(x^{2}+\sqrt{x}\right) d x = \int_{0}^{1}x^{2} d x + \int_{0}^{1}\sqrt{x} d x.\]
2Step 2: Evaluate the First Integral
Let's evaluate the first integral \(\int_{0}^{1}x^{2} d x\). We use the power rule for integration: \[\int x^{n} d x = \frac{x^{n+1}}{n+1} + C. \]Applying this rule, we get:\[\int_{0}^{1} x^{2} d x = \left.\frac{x^{3}}{3}\right|_{0}^{1}.\]Evaluating from 0 to 1, we get:\[\frac{1^{3}}{3} - \frac{0^{3}}{3} = \frac{1}{3}.\]
3Step 3: Evaluate the Second Integral
Now evaluate the second integral \(\int_{0}^{1}\sqrt{x} d x\). Recall that \(\sqrt{x} = x^{1/2}\). Using the power rule, we have:\[\int x^{1/2} d x = \frac{x^{1/2 + 1}}{1/2 + 1} + C = \frac{x^{3/2}}{3/2} + C.\]Simplifying, we get:\[\int_{0}^{1} \sqrt{x} d x = \left.\frac{2}{3}x^{3/2}\right|_{0}^{1}.\]Evaluating from 0 to 1, we obtain:\[\frac{2}{3}(1^{3/2}) - \frac{2}{3}(0^{3/2}) = \frac{2}{3}.\]
4Step 4: Combine the Results
Add the results of the two integrals from Steps 2 and 3:\[\frac{1}{3} + \frac{2}{3} = 1.\]Thus, the value of the original integral is 1.
Key Concepts
Integration TechniquesPower RuleDefinite IntegralsEvaluating Integrals
Integration Techniques
Integration techniques are strategies used to find the integral of a function. In the exercise, we encounter a technique called *separation* or *decomposition* of an integral. By splitting the integral \[ \int_{0}^{1}\left(x^{2}+\sqrt{x}\right) d x \]into \[ \int_{0}^{1}x^{2} d x + \int_{0}^{1}\sqrt{x} d x, \]we simplify the problem. Separating a complex integral into simpler parts is a common method that makes the process less daunting and calculations easier.
When you face an integral with multiple terms, consider breaking it down into smaller integrals. This technique allows a focused application of rules and can be especially helpful if the terms require different integration methods. Keep in mind that integration is not a one-size-fits-all process; breaking down integrals is just one of many techniques that you can use.
When you face an integral with multiple terms, consider breaking it down into smaller integrals. This technique allows a focused application of rules and can be especially helpful if the terms require different integration methods. Keep in mind that integration is not a one-size-fits-all process; breaking down integrals is just one of many techniques that you can use.
Power Rule
The power rule is one of the most fundamental methods in integral calculus. It allows us to integrate functions of the form \( x^n \), where \( n \) is any real number. The rule states: for any function \( x^n \), the integral is \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \]where \( C \) is the constant of integration.
In our exercise, this rule was applied twice. First for \( x^2 \) which yielded \[ \int_{0}^{1} x^2 \, dx = \left.\frac{x^3}{3}\right|_0^1 = \frac{1}{3}. \]Then, for \( \sqrt{x} = x^{1/2} \), applying the formula gives: \[ \int_{0}^{1} x^{1/2} \, dx = \left.\frac{x^{3/2}}{3/2}\right|_0^1 = \frac{2}{3}. \]Using the power rule, we can handle polynomials and expressions like \( \sqrt{x} \) effectively. This technique reduces more complicated expressions to manageable calculations.
In our exercise, this rule was applied twice. First for \( x^2 \) which yielded \[ \int_{0}^{1} x^2 \, dx = \left.\frac{x^3}{3}\right|_0^1 = \frac{1}{3}. \]Then, for \( \sqrt{x} = x^{1/2} \), applying the formula gives: \[ \int_{0}^{1} x^{1/2} \, dx = \left.\frac{x^{3/2}}{3/2}\right|_0^1 = \frac{2}{3}. \]Using the power rule, we can handle polynomials and expressions like \( \sqrt{x} \) effectively. This technique reduces more complicated expressions to manageable calculations.
Definite Integrals
Definite integrals calculate the *accumulated area* under a curve within given boundaries. For example, in the exercise, the integral \[ \int_{0}^{1}\left(x^{2}+\sqrt{x}\right) d x \]is evaluated between the limits \( 0 \) and \( 1 \).
The process involves finding the antiderivative first, then plugging the upper limit and subtracting the value at the lower limit. This evaluates the net area under the function from \( x=0 \) to \( x=1 \). In computational terms, the result provides the "total movement" or "accumulation" represented by the function over that interval.
For definite integrals, it is important to carefully manage these bounds to avoid potential errors with calculation.
The process involves finding the antiderivative first, then plugging the upper limit and subtracting the value at the lower limit. This evaluates the net area under the function from \( x=0 \) to \( x=1 \). In computational terms, the result provides the "total movement" or "accumulation" represented by the function over that interval.
For definite integrals, it is important to carefully manage these bounds to avoid potential errors with calculation.
Evaluating Integrals
Evaluating integrals means finding the numerical value for a definite integral. After applying rules like the power rule and techniques such as splitting the integral, computation becomes straightforward. For the exercise at hand, we evaluated \[ \int_{0}^{1} x^2 \, dx + \int_{0}^{1} x^{1/2} \, dx \]individually to get the respective areas. Then, to find a complete solution, we added them: \[ \frac{1}{3} + \frac{2}{3} = 1. \]
When evaluating, it's vital to undertake precise computation and simplify steps progressively, ensuring to account for all results accurately. This careful approach avoids errors and confirms the validity of the integration.
When evaluating, it's vital to undertake precise computation and simplify steps progressively, ensuring to account for all results accurately. This careful approach avoids errors and confirms the validity of the integration.
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