calculate ∫(from 8 to 8) (9x - 6)^2 dx, given the following. ∫(from 3 to 8) x^2 dx = 485/3 ∫(from 8 to 9)…
calculate ∫(from 8 to 8) (9x - 6)^2 dx, given the following. ∫(from 3 to 8) x^2 dx = 485/3 ∫(from 8 to 9) x^2 dx = 217/3 ∫(from 3 to 8) xdx = 55/2 ∫(from 8 to 8) (9x - 6)^2 dx = □
Answer
Explanation:
Step1: Expand the integrand
First, expand ((9x - 6)^2) using the formula ((a - b)^2=a^{2}-2ab + b^{2}). Here (a = 9x) and (b = 6), so ((9x - 6)^2=(9x)^{2}-2\times9x\times6 + 6^{2}=81x^{2}-108x + 36).
Step2: Use the integral - sum rule
(\int_{8}^{8}(9x - 6)^2dx=\int_{8}^{8}(81x^{2}-108x + 36)dx=81\int_{8}^{8}x^{2}dx-108\int_{8}^{8}xdx+36\int_{8}^{8}dx).
Step3: Recall the definite - integral property
For any definite integral (\int_{a}^{a}f(x)dx = 0). Since the upper and lower limits of integration are the same ((a = 8) in all three integrals), we have: (81\int_{8}^{8}x^{2}dx=81\times0 = 0), (-108\int_{8}^{8}xdx=-108\times0 = 0), and (36\int_{8}^{8}dx=36\times0 = 0).
Answer:
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