Evaluate the Integral

Problem

(∫_0^3)(|x−1|*d(x))

Solution

1. Identify the critical point where the expression inside the absolute value changes sign by setting x−1=0 which gives x=1

2. Split the integral into two parts based on the critical point within the interval [0,3]

(∫_0^3)(|x−1|*d(x))=(∫_0^1)(|x−1|*d(x))+(∫_1^3)(|x−1|*d(x))

3. Remove the absolute value bars by determining the sign of x−1 on each interval.

|x−1|=−(x−1)=1−x* for *0≤x≤1

|x−1|=x−1* for *1≤x≤3

4. Rewrite the integrals using the piecewise definitions.

(∫_0^3)(|x−1|*d(x))=(∫_0^1)((1−x)*d(x))+(∫_1^3)((x−1)*d(x))

5. Integrate each term using the power rule.

(∫_0^1)((1−x)*d(x))=[x−(x2)/2]10

(∫_1^3)((x−1)*d(x))=[(x2)/2−x]31

6. Evaluate the definite integrals at the boundaries.

[1−1/2]−[0−0]=1/2

[9/2−3]−[1/2−1]=3/2−(−1/2)=2

7. Sum the results of the two parts.

1/2+2=2.5

Final Answer

(∫_0^3)(|x−1|*d(x))=5/2

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