Problem 18 - Essential Calculus

Question

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

Step-by-Step Solution

Answer

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 1: 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].

Step 2: 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: \[ \int_{0}^{1} -(x^{2} - 4x + 3) dx , \int_{1}^{3} (x^{2} - 4x + 3) dx , \int_{3}^{4} -(x^{2} - 4x + 3) dx \]

Step 3: Evaluate Each Definite Integral

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

Step 4: 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.

Step 5: 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.