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Magazine Chapter 5: Problem 77
Find the average value over the given interval. $$ y=x^{n} ; \quad[0,1] $$
Short Answer
Expert verified
\( \frac{1}{n+1} \)
Step by step solution
01
Understand the Formula for Average Value
For a function defined on the interval \[a, b\], the average value of the function is given by: \(\frac{1}{b-a} \int_{a}^{b} f(x) \, dx\)
02
Identify the Function and Interval
Here, the given function is \(y = x^n\) and the interval is \([0, 1]\).
03
Set Up the Integral
Set up the integral for the function over the interval: \(\frac{1}{1-0} \int_{0}^{1} x^n \, dx\)
04
Evaluate the Integral
Evaluate the integral: \(\frac{1}{1-0} \int_{0}^{1} x^n \, dx = \int_{0}^{1} x^n \, dx \). To integrate \( x^n \), use the power rule: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \).
05
Apply the Limits of Integration
After finding the antiderivative, apply the limits of integration: \(\frac{x^{n+1}}{n+1} \bigg|_{0}^{1} \). This results in: \( \frac{1^{n+1}}{n+1} - \frac{0^{n+1}}{n+1} = \frac{1}{n+1} \)
06
Compute the Average Value
Since we already included the \( \frac{1}{b-a} \) factor in Step 3, we find that the average value of the function over the interval \([0, 1]\) is: \( \frac{1}{n+1} \)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
integration
Integration is a fundamental concept in calculus used to determine the area under a curve. This process helps find the sum of infinitely small areas, giving the whole area under the function's curve. The integral of a function over an interval gives you detailed information on the accumulated quantity.
One practical use of integration is in calculating the average value of a function over a specified interval. To perform integration:
One practical use of integration is in calculating the average value of a function over a specified interval. To perform integration:
- Identify the function to integrate
- Determine the bounds (or limits) of integration
- Find the antiderivative of the function
- Evaluate the antiderivative at the upper and lower bounds, then calculate the definite integral
- Use this definite integral result to get the desired quantity
power rule
The power rule is a shortcut in calculus for finding the derivative or integral of a power function. When dealing with expressions like \(x^n\), both differentiation and integration can be simplified using the power rule.
For integration, the power rule states: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \] where \(C\) is the constant of integration (only needed for indefinite integrals). This rule is essential for easily integrating functions of this form.
When integrating using the power rule:
For integration, the power rule states: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \] where \(C\) is the constant of integration (only needed for indefinite integrals). This rule is essential for easily integrating functions of this form.
When integrating using the power rule:
- Increase the exponent by 1
- Divide the result by the new exponent
- For definite integrals, substitute the upper and lower limits into the antiderivative and subtract the results
definite integral
A definite integral involves computing the integral within a specific interval \[a, b\]. Unlike an indefinite integral, the definite integral provides a numerical value representing the area under a curve between those limits. This value is critical in many real-world applications like physics, engineering, and biology.
To compute a definite integral:
To compute a definite integral:
- Set up the integral with the function and interval
- Find the antiderivative of the function
- Evaluate the antiderivative at the upper limit and lower limit of the interval
- Subtract the lower limit result from the upper limit result
Calculus for Life Sciences
Calculus plays a pivotal role in life sciences, including biology, medicine, and environmental science. It provides the mathematical foundation to model dynamic systems and understand changes over time.
Some applications include:
Some applications include:
- Modeling population growth using differential equations
- Understanding enzyme reactions through rate equations
- Calculating areas under curves to analyze biological data
- Computing average values to summarize biological functions
