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Magazine Chapter 7: Problem 9
\(1-80\) Evaluate the integral. $$\int_{1}^{3} r^{4} \ln r d r$$
Short Answer
Expert verified
\( \frac{243}{5} \ln 3 - \frac{242}{25} \).
Step by step solution
01
Identify the Integral Type
This is an integration problem involving the evaluation of the integral \( \int_{1}^{3} r^{4} \ln r \, dr \). The integrand is a product of a polynomial function \( r^4 \) and a logarithmic function \( \ln r \), which suggests that integration by parts will be a suitable technique.
02
Choose Functions for Integration by Parts
Recall the integration by parts formula: \[ \int u \, dv = uv - \int v \, du \]Let \( u = \ln r \), then \( du = \frac{1}{r} \, dr \). Let \( dv = r^4 \, dr \), then integrate to find \( v = \frac{r^5}{5} \).
03
Apply Integration by Parts Formula
Substitute the chosen \( u \), \( du \), \( v \), and \( dv \) into the integration by parts formula. This gives us:\[\int r^{4} \ln r \, dr = \frac{r^5}{5} \ln r \Bigg|_{1}^{3} - \int_{1}^{3} \left( \frac{r^5}{5} \right) \left( \frac{1}{r} \right) \, dr\]
04
Simplify and Evaluate the Remaining Integral
Simplify the remaining integral:\[\int_{1}^{3} \frac{r^5}{5r} \, dr = \int_{1}^{3} \frac{r^4}{5} \, dr = \frac{1}{5} \int_{1}^{3} r^4 \, dr\] Calculate \( \int r^4 \, dr \):\[\frac{1}{5} \cdot \left[ \frac{r^5}{5} \right]_{1}^{3} = \frac{1}{25} \left[ r^5 \right]_{1}^{3} = \frac{1}{25} (243 - 1) = \frac{1}{25} \times 242 = \frac{242}{25}\]
05
Evaluate the First Part of Integration by Parts
Evaluate \( \frac{r^5}{5} \ln r \Bigg|_{1}^{3} \):\[\frac{r^5}{5} \ln r \Bigg|_{1}^{3} = \left.\frac{243}{5} \ln 3 \right) - \left( \frac{1}{5} \ln 1 \right] = \frac{243}{5} \ln 3 - 0\] Thus, \( \frac{243}{5} \ln 3\) as \( \ln 1 = 0 \).
06
Combine Results
Combine the results of the integration by parts formula:\[\int_{1}^{3} r^{4} \ln r \, dr = \frac{243}{5} \ln 3 - \frac{242}{25}\] Simplify if needed, but this is the final expression for the evaluated integral.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Definite Integral
A definite integral is a fundamental concept in calculus that calculates the area under a curve between two points, called the limits of integration. Unlike indefinite integrals, which represent a family of functions, definite integrals provide a specific numerical value. In our problem, the integral is \[ \int_{1}^{3} r^{4} \ln r \, dr \]where 1 and 3 are the limits of integration.
Several key points about definite integrals include:
Several key points about definite integrals include:
- They involve both the process of integration and the evaluation of the resulting antiderivative at these limits.
- The notation encompasses the integral sign, integrand (expression to be integrated), and limits of integration.
- The Fundamental Theorem of Calculus connects differentiation and integration, showing that integrating and then differentiating brings us back to the original function.
- In our case, it calculates the net area between the curve defined by \( r^4 \ln r \) and the x-axis, from \( r = 1 \) to \( r = 3 \).
Logarithmic Function
Logarithmic functions are employed heavily in mathematics and sciences due to their unique properties. A logarithm, specifically the natural logarithm \(\ln r\), is the inverse operation of exponentiation with base \(e\), where \(e\) is approximately 2.718.
Some important characteristics are:
Some important characteristics are:
- The natural logarithm \( \ln r \) grows more slowly compared to polynomial functions as \( r \) increases.
- \( \ln 1 = 0 \), which simplifies calculations when evaluating definite integrals.
- Logarithmic functions can transform multiplicative relationships into additive ones; this property is used in integration techniques.
- For integration purposes, \( u = \ln r \) is often chosen because its derivative \( du = \frac{1}{r} \, dr \) is a simple rational expression that pairs well with polynomial factors in the integrand.
Polynomial Integration
Integrating polynomial expressions is a fundamental technique in calculus. Polynomials are expressions consisting of variables and coefficients, elevated to different power levels. In this problem, we encounter an integrand with a polynomial part \( r^4 \).
Key insights about polynomial integration include:
Key insights about polynomial integration include:
- The integration of a simple polynomial \( r^n \) follows the rule: \( \int r^n \, dr = \frac{r^{n+1}}{n+1} + C \) for \( n eq -1 \).
- The antiderivative of \( r^4 \) is \( \frac{r^5}{5} \), which aligns with general rules for integrating basic polynomial terms.
- Polynomial terms integrated individually adhere to linearity, allowing each term to be tackled separately.
- After determining the antiderivative, it must be evaluated within the specified limits of the definite integral to obtain a numerical solution.
