\int_1^7 (1-x)(x-7)dx
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Step 1: The problem asks to evaluate the definite integral: $$ \int_{1}^{7} (1-x)(x-7) dx $$ First, expand the integrand $(1-x)(x-7)$: $$ (1-x)(x-7) = 1(x-7) - x(x-7) $$ $$ = x - 7 - x^2 + 7x $$ $$ = -x^2 + 8x - 7 $$ So the integral becomes: $$ \int_{1}^{7} (-x^2 + Show more…
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