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Chapter 5: Problem 46

Evaluate. $$ \int_{0}^{1} \frac{12}{13} t^{2} d t $$

Short Answer

Expert verified
The value of the integral is \(\frac{4}{13}\).

Step by step solution

01

Identify the Integral

The integral to be evaluated is \(\frac{12}{13} \int_{0}^{1} t^{2} dt\). This is a definite integral over the interval [0, 1].
02

Factor Out the Constant Multiple

Since \(\frac{12}{13}\) is a constant, it can be factored out of the integral: \[ \frac{12}{13} \int_{0}^{1} t^{2} dt \]
03

Integrate the Function

Find the antiderivative of \(t^{2}\). The antiderivative of \(t^{2}\) is \(\frac{t^{3}}{3}\). So, \[ \frac{12}{13} \left[ \frac{t^{3}}{3} \right]_{0}^{1} \]
04

Evaluate at the Bounds

Evaluate the antiderivative at the upper and lower bounds. Substituting these bounds gives: \(\frac{12}{13} \left( \frac{1^{3}}{3} - \frac{0^{3}}{3} \right) = \frac{12}{13} \left( \frac{1}{3} \right) \)
05

Simplify the Result

Simplify the result: \[ \frac{12}{13} \cdot \frac{1}{3} = \frac{12}{39} = \frac{4}{13} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Antiderivative
An antiderivative is a function whose derivative equals the original function. In other words, if you have a function and you take its derivative, getting back the original function means you found its antiderivative.
For the function given in the problem, which is \( t^2 \), the antiderivative is found by applying the power rule:
To find the antiderivative of \( t^n \), use \( \frac{t^{n+1}}{n+1} \).
Applying this to \( t^2 \), we get \( \frac{t^{3}}{3} \).
So, the antiderivative of \( t^2 \) is \( \frac{t^{3}}{3} \).
This step is crucial as it transforms the function inside the integral into one we can easily evaluate.
Constant Multiple Rule
The constant multiple rule in calculus states that a constant factor can be moved outside an integral.
For instance, if you have an integral of the form \( \frac{12}{13} t^2 \), the constant \( \frac{12}{13} \) can be taken out:
\[ \frac{12}{13} \times \bigg(\text{integral of } t^2 \bigg) \]

This simplifies the computation as you only need to focus on finding the integral of the main function \( t^2 \).
Once you find the antiderivative, you multiply it by the constant multiple.
Upper and Lower Bounds
Definite integrals are evaluated over a specific interval, which is given by the upper and lower bounds.
In our example, the bounds are from 0 to 1. This means we need to evaluate our antiderivative at both the upper bound (1) and the lower bound (0).
Here's how it looks:
  • First, find the antiderivative: \( \frac{t^3}{3} \).
  • Evaluate this at the upper bound (1): \( \frac{1^3}{3} = \frac{1}{3} \).
  • Now evaluate it at the lower bound (0): \( \frac{0^3}{3} = 0 \).

The result of the definite integral is then the difference between these two values:
\[ \frac{1}{3} - 0 = \frac{1}{3} \]
That's the contribution of the function over the interval from 0 to 1.
Integral Simplification
After finding the integral and evaluating it at the bounds, we need to simplify the result.
We found the antiderivative and evaluated it to get \( \frac{1}{3} \).
The problem had a constant multiple \( \frac{12}{13} \), so we multiply it by our result:
\[ \frac{12}{13} \times \frac{1}{3} = \frac{12}{39} \]

To simplify \( \frac{12}{39} \), notice that both numerator and denominator have a common factor of 3:
\[ \frac{12 \div 3}{39 \div 3} = \frac{4}{13} \]

Thus, the final simplified result of the given definite integral is \( \frac{4}{13} \). This simplification helps in presenting the most reduced and simplified form of the answer.

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