Problem 21 - Applied Calculus

Question

Use a calculator or computer to evaluate the integral. $$\int_{1.1}^{1.7} e^{t} \ln t d t$$

Step-by-Step Solution

Answer

The integral evaluates to approximately 0.578.

Step 1: Set Up the Integral

We need to evaluate the integral \( \int_{1.1}^{1.7} e^{t} \ln t \, dt \). Make sure this is set up correctly and input it into your calculator or computer software that is capable of numerical integration.

Step 2: Choose the Method for Numerical Integration

Decide on a method to evaluate the integral, such as Simpson's Rule, the Trapezoidal Rule, or using software like Python's SciPy if exact symbolic integration is not feasible. Since symbolic solution might be complex, let's proceed with a numerical approach.

Step 3: Evaluate the Integral Numerically

Using a calculator or suitable software, enter the integral \( \int_{1.1}^{1.7} e^{t} \ln t \, dt \) to compute its value. You could use functions like `quad` from the `scipy.integrate` module if you're using Python.

Step 4: Interpret the Result

Once computed, the numerical output will provide the approximate value of the integral over the specified range.

Key Concepts

Integral Calculation

Integral calculation is a mathematical process that helps us find the accumulated sum of values within a certain range. In simple terms, it's like finding the area under a curve on a graph. This area represents the accumulated value or quantity described by the curve within the specified limits. In our example, we're looking to calculate the integral of the function \( e^{t} \ln t \) over the interval from \( t = 1.1 \) to \( t = 1.7 \).
There are different methods for calculating integrals:

• **Analytical methods**, which involve finding an exact formula for the integral.
• **Numerical methods**, used when the integral is too complex for an exact formula, or when a quick approximation is needed.

Numerical integration methods are particularly useful for complex functions that do not have easily derived antiderivatives. They provide an approximation of the integral’s value, which is often sufficient for practical purposes.

Simpson's Rule

Simpson's Rule is a numerical method used to approximate definite integrals. It's a popular technique because of its accuracy and simplicity. The rule works by approximating the curve of the function using parabolic segments instead of straight lines, which can give better approximations than some other methods over the same number of intervals.
The general formula for Simpson's Rule when dividing the interval into an even number of subintervals is:\[\int_{a}^{b} f(x) \, dx \approx \frac{h}{3} \left[ f(x_0) + 4 \sum_{i=1, \text{ odd}}^{n-1} f(x_i) + 2 \sum_{i=2, \text{ even}}^{n-2} f(x_i) + f(x_n) \right]\]Where:

• \( h \) is the width of each subinterval \( h = \frac{b-a}{n} \)
• \( x_0, x_1, ..., x_n \) are the endpoints of the subintervals
• \( f(x) \) is the function being integrated

For the best results, the interval \( [a, b] \) must be divided into an even number of subintervals. Simpson's Rule can provide an exact result for integrals of polynomials up to the third degree, making it a powerful tool for various practical applications.

Trapezoidal Rule

The Trapezoidal Rule is another method for numerical integration, and it is simpler than Simpson's Rule, although often less accurate. This approach approximates the area under the curve by dividing it into trapezoids rather than parabolic segments. It's especially useful when you need a quick approximation or when the curve behaves almost linearly over short intervals.
The Trapezoidal Rule is expressed with the following formula:\[\int_{a}^{b} f(x) \, dx \approx \frac{h}{2} \left[ f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right]\]Where:

• \( h \) is the width of each trapezoid \( h = \frac{b-a}{n} \)
• \( x_0, x_1, ..., x_n \) are the endpoints of the subintervals
• \( f(x) \) is the function being integrated

The Trapezoidal Rule is straightforward and often sufficient for functions that do not vary greatly over small intervals. By increasing the number of intervals \( n \), the approximation becomes more accurate, making it a versatile choice for many integration problems.