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Chapter 4: Problem 54

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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.
04

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.

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Most popular questions from this chapter

The area \(A\) between the graph of the function \(g(t)=4-4 / t^{2}\) and the \(t\) -axis over the interval \([1, x]\) is \(A(x)=\int_{1}^{x}\left(4-\frac{4}{t^{2}}\right) d t\) (a) Find the horizontal asymptote of the graph of \(g\). (b) Integrate to find \(A\) as a function of \(x\). Does the graph of \(A\) have a horizontal asymptote? Explain.

From the vertex \((0, c)\) of the catenary \(y=c \cosh (x / c)\) a line \(L\) is drawn perpendicular to the tangent to the catenary at a point \(P\). Prove that the length of \(L\) intercepted by the axes is equal to the ordinate \(y\) of the point \(P\).

Use the Second Fundamental Theorem of Calculus to find \(F^{\prime}(x)\). $$ F(x)=\int_{1}^{x} \frac{t^{2}}{t^{2}+1} d t $$

Use the Second Fundamental Theorem of Calculus to find \(F^{\prime}(x)\). $$ F(x)=\int_{0}^{x} t \cos t d t $$

A model for a power cable suspended between two towers is given. (a) Graph the model, (b) find the heights of the cable at the towers and at the midpoint between the towers, and (c) find the slope of the model at the point where the cable meets the right-hand tower. \(y=18+25 \cosh \frac{x}{25}, \quad-25 \leq x \leq 25\)
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