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Chapter 5: Problem 73

Use geometry to evaluate the following integrals. $$\int_{1}^{6}|2 x-4| d x$$

Short Answer

Expert verified
Answer: The value of the definite integral is 17.

Step by step solution

01

Identify the zero point of the function

First, let's identify the point where the value inside the absolute value function becomes zero. This occurs when 2x-4=0. Solving for x: $$2x-4=0$$ $$2x=4$$ $$x=2$$ So, we have a zero point at x = 2. This means that the function |2x-4| changes its behavior at x = 2.
02

Split the integral function into two parts

Now that we know the zero point, we can split the integral into two parts: one where the function (2x-4) is negative, and the other where it is non-negative. This happens at x=1 to x=2 and x=2 to x=6, respectively. $$\int_{1}^{6}|2x-4| dx = \int_{1}^{2}-(2x-4) dx + \int_{2}^{6}(2x-4) dx$$
03

Evaluate each part

Now, we can evaluate the two parts of the integral separately. $$\int_{1}^{2}-(2x-4) dx = -\left[ x^2-4x\right]_{1}^{2} = -(4-8) - (1-4)= 4-3 =1$$ $$\int_{2}^{6}(2x-4) dx = \left[ x^2-4x\right]_{2}^{6} = (36-24) - (4-8)= 12+4 =16$$
04

Add the two parts of the integral

Finally, we add the results of the two parts of the integral: $$\int_{1}^{6}|2x-4| dx = 1+16 = 17$$ So, the value of the integral is 17.

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Most popular questions from this chapter

Consider the function \(f(x)=x^{2}-4 x\) a. Graph \(f\) on the interval \(x \geq 0\) b. For what value of \(b>0\) is \(\int_{0}^{b} f(x) d x=0 ?\) c. In general, for the function \(f(x)=x^{2}-a x,\) where \(a>0,\) for what value of \(b>0\) (as a function of \(a\) ) is \(\int_{0}^{b} f(x) d x=0 ?\)

Assume that the linear function \(f(x)=m x+c\) is positive on the interval \([a, b] .\) Prove that the midpoint Riemann sum with any value of \(n\) gives the exact area of the region between the graph of \(f\) and the \(x\) -axis on \([a, b]\).

Looking ahead: Integrals of \(\tan x\) and \(\cot x\) a. Use a change of variables to show that $$\int \tan x d x=-\ln |\cos x|+C=\ln |\sec x|+C$$ b. Show that $$\int \cot x d x=\ln |\sin x|+C$$.

Average value of sine functions Use a graphing utility to verify that the functions \(f(x)=\sin k x\) have a period of \(2 \pi / k,\) where \(k=1,2,3, \ldots . .\) Equivalently, the first "hump" of \(f(x)=\sin k x\) occurs on the interval \([0, \pi / k] .\) Verify that the average value of the first hump of \(f(x)=\sin k x\) is independent of \(k .\) What is the average value?

Consider the function g. which is given in terms of a definite integral with a variable upper limit. a. Graph the integrand. b. Calculate \(g^{\prime}(x)\) c. Graph g, showing all your work and reasoning. $$g(x)=\int_{0}^{x} \sin \left(\pi t^{2}\right) d t \text { (a Fresnel integral) }$$
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