Evaluate the Integral

Problem

(∫_−3^4)(|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 where the expression inside the absolute value changes sign, which is x=0

(∫_−3^4)(|x|*d(x))=(∫_−3^0)(−x*d(x))+(∫_0^4)(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 for both intervals.

[−(x2)/2]0(−3)=−0/2−(−((−3)2)/2)=0−(−9/2)=9/2

[(x2)/2]40=4/2−0/2=16/2−0=8

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

9/2+8=9/2+16/2=25/2

Final Answer

(∫_−3^4)(|x|*d(x))=25/2

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