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Chapter 4: Problem 18

Evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. $$ \int_{0}^{4}\left|x^{2}-4 x+3\right| d x $$

Short Answer

Expert verified
The solution to the definite integral \(\int_{0}^{4}\left|x^{2}-4 x+3\right| dx\) is 2. To verify this, use a graphing utility to plot the absolute function and calculate the area under the curve.

Step by step solution

01

Partition the Function into Distinct Intervals

The expression within the absolute value brackets \(x^{2}-4x+3\) can be factored into \((x-1)(x-3)\). Then, set each factor equal to zero to determine the x-values of 1 and 3 where the expression inside the absolute value changes the sign. As such, the integral can be partitioned into three intervals: [0,1], [1,3], and [3,4].
02

Set up the Integral for Each Interval

Set up an integral for each interval according to the sign of the expression inside the absolute value in that particular interval: \[ \int_{0}^{1} -(x^{2} - 4x + 3) dx , \int_{1}^{3} (x^{2} - 4x + 3) dx , \int_{3}^{4} -(x^{2} - 4x + 3) dx \]
03

Evaluate Each Definite Integral

For each of the definite integrals, apply the power rule for antidifferentiation: \( -[\frac{1}{3}x^{3} - 2x^{2} + 3x]_{0}^{1} , [\frac{1}{3}x^{3} - 2x^{2} + 3x]_{1}^{3} , -[\frac{1}{3}x^{3} - 2x^{2} + 3x]_{3}^{4} \)
04

Calculate the Results of Each Definite Integral

Substitute the limits of the integral into the antiderivative and subtract the results. Afterwards, sum all the results.
05

Verify With a Graphing Utility

Use a graphing utility to plot the absolute function and calculate the area under the curve, ensuring it agrees with the result obtained.

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