Problem 63 - Calculus and its applications

Question

Use a graphing calculator to evaluate $$ \int_{1}^{10} x^{5} \ln x d x $$

Step-by-Step Solution

Answer

Use a graphing calculator to find the numerical value of \( \int_{1}^{10} x^{5} \ln x \, dx \).

Step 1: Set up the Integral for Evaluation

We need to evaluate the definite integral \( \int_{1}^{10} x^{5} \ln x \, dx \). This integral involves a product of a polynomial \( x^5 \) and the natural logarithm \( \ln x \), which can be complex to solve manually without integration by parts.

Step 2: Use a Graphing Calculator

Use a graphing calculator that supports definite integrals to compute \( \int_{1}^{10} x^{5} \ln x \, dx \). Enter the function \( x^5 \ln x \) and set the limits of integration from 1 to 10.

Step 3: Read the Calculator Output

After entering the correct function and limits, the calculator provides a numerical approximation for the integral over the interval \([1, 10]\). Note down the value displayed on the screen.

Key Concepts

Graphing Calculator

A graphing calculator is an essential tool that can graph functions and perform complex calculations such as definite integrals. To use one for evaluating a definite integral, like \( \int_{1}^{10} x^{5} \ln x \, dx \), follow these steps:

• First, ensure your calculator is capable of handling definite integrals. Most advanced calculators provide this feature under the calculus or math functions menu.
• Input the integral function. In this case, you enter \( x^{5} \ln x \) and set your bounds from 1 to 10.
• After inserting the function and limits, execute the calculation. The calculator uses numerical methods to estimate the result, providing a quick solution that would be cumbersome to compute manually.

When you receive the output, it's a numerical approximation of the integral, which is usually accurate to several decimal places. This approximation is useful for validating manual calculations or when an exact algebraic solution is not necessary.

Integration by Parts

Integration by parts is an essential technique in calculus used to integrate the product of two functions. It's based on the product rule of derivatives. The formula is:
\[\int u \cdot dv = uv - \int v \cdot du\]For the integral \( \int_{1}^{10} x^{5} \ln x \, dx \), you apply this technique because it involves both a polynomial and a logarithmic function. Selecting:

• \( u = \ln x \) allows \( du = \frac{1}{x} \, dx \)
• \( dv = x^5 \, dx \) gives \( v = \frac{x^6}{6} \)

You substitute these into the formula, resulting in:\[\left. \left( \ln x \cdot \frac{x^6}{6} \right) \right|_1^{10} - \int_{1}^{10} \frac{x^6}{6} \cdot \frac{1}{x} \, dx\]Calculating this manually poses challenges. This is why people often use graphing calculators to simplify the process, especially when exact precision isn't the primary concern.

Numerical Approximation

Numerical approximation is crucial when dealing with problems that are complex or impossible to solve analytically. For definite integrals, approximation methods, like those used by graphing calculators, provide practical results.
Approximation methods used include:

• **Trapezoidal Rule**: This method estimates the area under a curve by dividing it into a series of trapezoids and calculating their total area.
• **Simpson's Rule**: It improves on the trapezoidal rule by using parabolic arcs instead of straight lines, enhancing accuracy.

In graphing calculators, these methods are automated, allowing them to rapidly compute the approximate value of a definite integral. The value obtained is an estimation with high accuracy and can be used for further analysis or decision-making.
For the integral \( \int_{1}^{10} x^{5} \ln x \, dx \), using a graphing calculator simplifies finding the approximate numerical result, making it accessible for students and engineers alike.