Questions: Question Use a Computer Algebra System (CAS) to calculate the area below f(x) = 4x / (x^4 - 1), above the x-axis, and between x=3 and x=4, as shown by the shaded area on the graph below. Round your answer to three decimal places, if necessary.

Question
Use a Computer Algebra System (CAS) to calculate the area below f(x) = 4x / (x^4 - 1), above the x-axis, and between x=3 and x=4, as shown by the shaded area on the graph below.

Round your answer to three decimal places, if necessary.

Transcript text: Question Use a Computer Algebra System (CAS) to calculate the area below $f(x)=\frac{4 x}{x^{4}-1}$, above the $x$-axis, and between $x=3$ and $x=4$, as shown by the shaded area on the graph below. Round your answer to three decimal places, if necessary.

Solution

Solution Steps

To find the area under the curve $ f(x) = \frac{4x}{x^4 - 1} $ from $ x = 3 $ to $ x = 4 $, we need to compute the definite integral of the function over this interval. This can be done using a Computer Algebra System (CAS) or a numerical integration method in Python.

Step 1: Define the Function

We start with the function $ f(x) = \frac{4x}{x^4 - 1} $. This function describes the curve for which we want to find the area under it between the specified limits.

Step 2: Set Up the Integral

To find the area under the curve above the $ x $-axis from $ x = 3 $ to $ x = 4 $, we set up the definite integral: \[ A = \int_{3}^{4} f(x) \, dx = \int_{3}^{4} \frac{4x}{x^4 - 1} \, dx \]

Step 3: Calculate the Integral

Evaluating the definite integral gives us the area $ A $. After performing the calculation, we find: \[ A \approx 0.098 \]