Evaluate the Integral

Problem

(∫_−1^2)(|x|*d(x))

Solution

1. Identify the definition of the absolute value function |x| which changes behavior at x=0

|x|=−x* if *x<0

|x|=x* if *x≥0

2. Split the integral into two parts based on the interval of integration [−1,2] using the point x=0

(∫_−1^2)(|x|*d(x))=(∫_−1^0)(−x*d(x))+(∫_0^2)(x*d(x))

3. Find the antiderivatives for each part of the split integral.

(∫_^)(−x*d(x))=−(x2)/2

(∫_^)(x*d(x))=(x2)/2

4. Apply the Fundamental Theorem of Calculus by evaluating the antiderivatives at the upper and lower limits.

(∫_−1^0)(−x*d(x))=[−(x2)/2]0(−1)=(0)−(−((−1)2)/2)=1/2

(∫_0^2)(x*d(x))=[(x2)/2]20=(2/2)−(0)=2

5. Sum the results of the two definite integrals to find the total area.

1/2+2=5/2

Final Answer

(∫_−1^2)(|x|*d(x))=5/2

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