Problem 54 - Essential Calculus

Question

Find the area of the region bounded by the graphs of the equations. Use a graphing utility to verify your result. $$ y=2 x-\tan (0.3 x), x=1, x=4, y=0 $$

Step-by-Step Solution

Answer

The answer is the absolute value of the integral of the given function from \(x=1\) to \(x=4\), which can be computed numerically due to the complexity of the function.

Step 1: Plot the Functions

Use a graphing utility to plot \(y=2x-\tan(0.3x)\), \(x=1\), \(x=4\), and \(y=0\) on the same graph. Notice that \(y=2x-\tan(0.3x)\) is an oscillating function that crosses the x-axis at various points between \(x=1\) and \(x=4\). The area between the curve of this function and the x-axis in between these vertical lines is the area we want to find.

Step 2: Determine the Integral

The area under the curve of the function \(y=2x-\tan(0.3x)\) from \(x=1\) to \(x=4\) is given by \(\int_{1}^{4} |2x-\tan(0.3x)| dx\). Because we are interested in the area, we consider the absolute value of the integrated function. So the negative 'humps' that are below the x-axis will be considered as positive area.

Step 3: Calculate the Integral for the Area

Calculate the integral by considering the absolute value of the function to get the overall area under the curve between \(x=1\) and \(x=4\). This integral may not have an elementary expression and can be computed numerically using numerical integration techniques or software like Mathematica or MATLAB.

Step 4: Verify the Result

Use the shading feature on your graphing utility to verify the result. This will shade the area under the curve of the equation between \(x=1\) and \(x=4\), helping you visually verify if your integral calculation is likely correct.