Graph the integrand and use geometry to evaluate the integral.

Question

BlackTom AI · Accepted Answer

We start by identifying the integrand and limits: the integral is ∫ from -5 to 5 of (|x| + 8) dx. This can be split into two simpler integrals: ∫_{-5}^{5} |x| dx plus ∫_{-5}^{5} 8 dx. 
Option 1: 105. This is the correct total, since ∫_{-5}^{5} |x| dx equals twice the area under y = x from 0 to 5, i.e., 2 * ∫_{0}^{5} x dx = 2 * (1/2 * 25) = 25. And ∫_{-5}^{5} 8 dx = 8 * (5 - (-5)) = 8 * 10 = 80. Adding them gives 25 + 80 = 105.
Option 2: 65. This would imply a smaller contribution from either the |x| part or the constant part, but as shown, the |x| part is 25 and the constant part is 80, summing to 105, not 65. A miscalculation might arise from forgetting symmetry or incorrectly integrating |x| or the constant.
Option 3: 185. This overestimates the result by at least 80 more than the correct 105. It could come from double-counting the constant 8 over the interval or incorrectly integrating |x| as 40 instead of 25, which would yield 120, still not 185, so it’s inconsistent with the actual areas.
Option 4: 130. This is 25 more than 105. It might reflect adding only half of the constant part (80) plus the full |x| part (25) and misplacing an extra 25 somewhere, but the correct total from the decomposition is exactly 25 + 80 = 105, so 130 is not consistent with the exact areas involved.
In summary, the integral splits neatly into the sum of the two simple integrals, yielding 105, and the other options arise from common arithmetic or conceptual mistakes about symmetry, the absolute value, or overcounting the constant term.

MATH2415W16 Quiz 4 Dec 16 (W16)- Requires Respondus LockDown Browser

Graph the integrand and use geometry to evaluate the integral.

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