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Magazine Chapter 5: Problem 28
Evaluate the integral. $$ \int_{0}^{4}(4-t) \sqrt{t} d t $$
Short Answer
Expert verified
The value of the integral is \( \frac{128}{15} \).
Step by step solution
01
Understand the Integral
We need to evaluate the definite integral \( \int_{0}^{4}(4-t)\sqrt{t} \, dt \). This integral involves a product of two functions, \( 4-t \) and \( \sqrt{t} \), and requires integration with respect to \( t \) from 0 to 4.
02
Distribute and Simplify
To simplify our work, distribute \( \sqrt{t} \) across \( (4-t) \) inside the integral: \( \int_{0}^{4} (4\sqrt{t} - t\sqrt{t}) \, dt \). This gives us two separate integrals: \( \int_{0}^{4} 4\sqrt{t} \, dt - \int_{0}^{4} t\sqrt{t} \, dt \).
03
Set Up Each Integral Separately
First, consider the integral \( \int_{0}^{4} 4\sqrt{t} \, dt \), which can be written as \( 4 \int_{0}^{4} t^{1/2} \, dt \). Second, consider \( \int_{0}^{4} t\sqrt{t} \, dt \), which simplifies to \( \int_{0}^{4} t^{3/2} \, dt \).
04
Integrate \( 4 \int_{0}^{4} t^{1/2} \, dt \)
The integral \( 4 \int_{0}^{4} t^{1/2} \, dt \) is a power function. Use the power rule for integration: \( \int t^{n} \, dt = \frac{t^{n+1}}{n+1} \). Thus, \( \int t^{1/2} \, dt = \frac{t^{3/2}}{3/2} = \frac{2}{3}t^{3/2} \). Evaluating from 0 to 4: \( 4 \times \left[ \frac{2}{3} \left(4^{3/2} - 0 \right) \right] = \frac{8}{3} \times 8 = \frac{64}{3} \).
05
Integrate \( \int_{0}^{4} t^{3/2} \, dt \)
Similarly, integrate \( \int_{0}^{4} t^{3/2} \, dt \) using the power rule: \( \int t^{3/2} \, dt = \frac{t^{5/2}}{5/2} = \frac{2}{5} t^{5/2} \). Evaluating from 0 to 4: \( \frac{2}{5} \left(4^{5/2} - 0 \right) = \frac{2}{5} \times 32 = \frac{64}{5} \).
06
Subtract the Results and Simplify
Subtract the result of the second integral from the first: \( \frac{64}{3} - \frac{64}{5} \). To do this, find a common denominator, which is 15. Convert each fraction: \( \frac{64}{3} = \frac{320}{15} \) and \( \frac{64}{5} = \frac{192}{15} \). Now subtract: \( \frac{320}{15} - \frac{192}{15} = \frac{128}{15} \).
07
Final Step: Conclusion
The value of the definite integral \( \int_{0}^{4} (4-t)\sqrt{t} \, dt \) is \( \frac{128}{15} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Power Rule for Integration
One of the foundational concepts in calculus is the **Power Rule for Integration**. This rule makes it straightforward to integrate functions that are power of a variable. When you come across a term like \( t^n \), the power rule states that the integral of this term with respect to \( t \) is given by:\[\int t^n \, dt = \frac{t^{n+1}}{n+1} + C\]where \( C \) is the constant of integration, which appears in indefinite integrals. However, in definite integrals like the one we have, the constant is not included as we evaluate over a specific interval.For our original exercise:
- The first part, \( \int_{0}^{4} t^{1/2} \, dt \), uses the power rule to find that \( \int t^{1/2} \, dt = \frac{2}{3}t^{3/2} \).
- The second part, \( \int_{0}^{4} t^{3/2} \, dt \), applies the power rule to find \( \int t^{3/2} \, dt = \frac{2}{5} t^{5/2} \).
Integration Techniques
Solving integrals often requires more than one technique. In our example, we simplify the integrand by using algebraic manipulation. Here's how we can efficiently employ such **Integration Techniques**:
- **Distribution:** In our problem, the expression \( (4 - t)\sqrt{t} \) was distributed as \( 4\sqrt{t} - t\sqrt{t} \). This breaks down a complicated integral into simpler parts.
- **Simplification:** Each of these parts was simplified into a form where the power rule is applicable: \( 4 \int_{0}^{4} t^{1/2} \, dt \) and \( \int_{0}^{4} t^{3/2} \, dt \).
Evaluating Definite Integrals
After simplifying and integrating, the next step is **Evaluating Definite Integrals** over a specified range. This involves applying limits of integration, which is the process of substituting the upper and lower bounds into your antiderivatives and then subtracting the results.For our task:
- The value of \(\frac{2}{3}t^{3/2}\) was computed at the boundaries \(0\) and \(4\), resulting in \( \frac{64}{3} \).
- Similarly, \(\frac{2}{5}t^{5/2}\) was evaluated from \(0\) to \(4\), resulting in \( \frac{64}{5} \).
