Evaluate the Integral

Problem

(∫_0^5)(|2*x−5|*d(x))

Solution

1. Identify the critical point where the expression inside the absolute value changes sign by setting 2*x−5=0

2*x−5=0

x=2.5

2. Split the integral into two parts based on the critical point x=2.5 within the interval [0,5]

(∫_0^5)(|2*x−5|*d(x))=(∫_0^2.5)(−(2*x−5)*d(x))+(∫_2.5^5)((2*x−5)*d(x))

3. Simplify the integrands by distributing the negative sign in the first integral.

(∫_0^2.5)((−2*x+5)*d(x))+(∫_2.5^5)((2*x−5)*d(x))

4. Find the antiderivatives for both expressions.

[−x2+5*x]2.50+[x2−5*x]52.5

5. Evaluate the first definite integral at the boundaries 0 and 2.5

(−(2.5)2+5*(2.5))−(−(0)2+5*(0))

−6.25+12.5=6.25

6. Evaluate the second definite integral at the boundaries 2.5 and 5

((5)2−5*(5))−((2.5)2−5*(2.5))

(25−25)−(6.25−12.5)=0−(−6.25)=6.25

7. Sum the results of the two integrals to find the total area.

6.25+6.25=12.5

Final Answer

(∫_0^5)(|2*x−5|*d(x))=12.5

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