Evaluate the Integral

Problem

(∫_−1^1)(|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.

(∫_−1^1)(|x|*d(x))=(∫_−1^0)(−x*d(x))+(∫_0^1)(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. Evaluate the definite integrals by applying the Fundamental Theorem of Calculus.

[−(x2)/2]0(−1)=−0/2−(−((−1)2)/2)=0−(−1/2)=1/2

[(x2)/2]10=1/2−0/2=1/2−0=1/2

5. Sum the results of the two parts to find the total area.

1/2+1/2=1

Final Answer

(∫_−1^1)(|x|*d(x))=1

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