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Magazine Chapter 5: Problem 6
\(1-30=\) Evaluate the integral. $$\int_{-1}^{1} t(1-t)^{2} d t$$
Short Answer
Expert verified
The integral evaluates to \(-\frac{4}{3}\).
Step by step solution
01
Expand the Integrand
First, expand the expression \(t(1-t)^2\) using the binomial theorem. This gives:\[t(1-t)^2 = t(1 - 2t + t^2) = t - 2t^2 + t^3.\]
02
Set Up the Integral
Now, substitute the expanded expression back into the integral:\[\int_{-1}^{1} (t - 2t^2 + t^3) \, dt.\]
03
Integrate Term by Term
Integrate each term separately:- For \(t\), the integral is \(\frac{t^2}{2}\).- For \(-2t^2\), the integral is \(-\frac{2t^3}{3}\).- For \(t^3\), the integral is \(\frac{t^4}{4}\).Thus, the integrated expression is:\[\frac{t^2}{2} - \frac{2t^3}{3} + \frac{t^4}{4}.\]
04
Evaluate the Definite Integral
Apply the limits \([-1, 1]\) to the integrated expression:First evaluate at \(t = 1\):\[\left(\frac{1^2}{2} - \frac{2(1)^3}{3} + \frac{1^4}{4}\right) = \frac{1}{2} - \frac{2}{3} + \frac{1}{4}.\]Now, evaluate at \(t = -1\):\[\left(\frac{(-1)^2}{2} - \frac{2(-1)^3}{3} + \frac{(-1)^4}{4}\right) = \frac{1}{2} + \frac{2}{3} + \frac{1}{4}.\]
05
Simplify the Results
Simplify both evaluations:- For \(t = 1\), compute: \[\frac{1}{2} - \frac{2}{3} + \frac{1}{4} = \frac{6}{12} - \frac{8}{12} + \frac{3}{12} = \frac{1}{12}.\]- For \(t = -1\), compute: \[\frac{1}{2} + \frac{2}{3} + \frac{1}{4} = \frac{6}{12} + \frac{8}{12} + \frac{3}{12} = \frac{17}{12}.\]
06
Subtract to Find the Result
Subtract the evaluation at \(t = -1\) from the evaluation at \(t = 1\):\[\frac{1}{12} - \frac{17}{12} = -\frac{16}{12} = -\frac{4}{3}.\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binomial Theorem
The Binomial Theorem is a fundamental principle in mathematics that allows us to expand expressions of the form \((a + b)^n\). It provides a way to rewrite polynomial expressions using a series of terms involving binomial coefficients.
In this exercise, we use the Binomial Theorem to expand the polynomial \((1 - t)^2\).This is utilized since our expression inside the integral is \(t(1-t)^2\).
By expanding \((1 - t)^2\), we get \(1 - 2t + t^2\).Multiplying each of these terms by \(t\) gives us the expanded form \(t - 2t^2 + t^3\).
This expansion is crucial for integrating each term individually in polynomial integration.
In this exercise, we use the Binomial Theorem to expand the polynomial \((1 - t)^2\).This is utilized since our expression inside the integral is \(t(1-t)^2\).
By expanding \((1 - t)^2\), we get \(1 - 2t + t^2\).Multiplying each of these terms by \(t\) gives us the expanded form \(t - 2t^2 + t^3\).
This expansion is crucial for integrating each term individually in polynomial integration.
Polynomial Integration
Integration is a mathematical process used to find areas under curves or accumulate quantities over intervals. When it comes to polynomial integration, each term of a polynomial is integrated separately.
In this particular problem, after expansion using the Binomial Theorem, the polynomial \(t - 2t^2 + t^3\) is integrated.
Here's how each term is integrated:
In this particular problem, after expansion using the Binomial Theorem, the polynomial \(t - 2t^2 + t^3\) is integrated.
Here's how each term is integrated:
- For \(t\), the integral is calculated as \(\frac{t^2}{2}\).This uses the rule: integrate \(t^n\) to get \(\frac{t^{n+1}}{n+1}\).
- For \(-2t^2\), we integrate it to\(-\frac{2t^3}{3}\).
- For \(t^3\), we obtain\(\frac{t^4}{4}\).
Definite Integration Limits
Definite integrals yield a numerical value that represents the area under the curve between two specified limits. Thus, it's different from indefinite integrals that result in a general antiderivative function plus a constant.
For the problem at hand, the limits provided are from \(-1\) to \(1\). Therefore, to compute the definite integral, you need to evaluate the integrated expression at these bounds.
After integration, the expression is \(\frac{t^2}{2} - \frac{2t^3}{3} + \frac{t^4}{4}\).To find the definite integral:
Utilizing these steps ensures that the calculation of definite integrals is thorough and correct.
For the problem at hand, the limits provided are from \(-1\) to \(1\). Therefore, to compute the definite integral, you need to evaluate the integrated expression at these bounds.
After integration, the expression is \(\frac{t^2}{2} - \frac{2t^3}{3} + \frac{t^4}{4}\).To find the definite integral:
- Evaluate the expression at the upper limit, \(t=1\).
- Then, evaluate the expression at the lower limit, \(t=-1\).
- Subtract the results: \(\text{value at } t=1 - \text{value at } t=-1\).
Utilizing these steps ensures that the calculation of definite integrals is thorough and correct.
