Evaluate the Integral

Problem

(∫_−2^2)(|x|*d(x))

Solution

1. Identify the definition of the absolute value function |x| which is x when x≥0 and −x when x<0

2. Split the integral at the point x=0 where the behavior of the absolute value changes.

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

3. Find the antiderivatives for both parts of the expression.

(∫_^)(−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.

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

2+2=4

Final Answer

(∫_−2^2)(|x|*d(x))=4

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