Porisma

Worked solution

Compare a modulus integral with signed integrals

Without using a calculator, identify which of the following definite integral(s) has a different value. Refer to the sketch from Example 9: (a) ∫−20∣x2−4∣ dx\int_{-2}^0|x^2-4|\,dx, (b) ∫−20(x2−4) dx\int_{-2}^0(x^2-4)\,dx, (c) −∫−20(x2−4) dx-\int_{-2}^0(x^2-4)\,dx. You can check your own answer using the graphing calculator.

CJC · Lecture · Self-Practice 3

Compare signs first

On [−2,0][-2,0], x2−4≤0x^2-4\le0. Consequently ∣x2−4∣=−(x2−4)|x^2-4|=-(x^2-4). Therefore (a) equals (c), while (b) has the opposite sign and is the different value.

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Optional exact verification

∫−20(x2−4) dx=[x3/3−4x]−20=0−(−8/3+8)=−16/3\int_{-2}^0(x^2-4)\,dx=[x^3/3-4x]_{-2}^0=0-(-8/3+8)=-16/3. Thus (a) and (c) are 16/316/3 and (b) is −16/3-16/3. This verification is independently authored; neither source prints a final answer for Self-Practice 3.

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Common mistakes

The modulus of the integrand changes pointwise signs; here it equals the negated integrand because the original stays nonpositive.

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