Question Detail

To solve this problem, we first evaluate the indefinite integral ∫ (2x-6)2 dx and then apply the limits of integration.Expand the integrand: (2x-6)2 = (2x)2 - 2(2x)(6) + 62 = 4x2 - 24x + 36.Now, integrate term by term:∫ (4x2 - 24x + 36) dx = (4x3/3) - (24x2/2) + 36x + C = (4/3)x3 - 12x2 + 36x + C.Next, apply the definite integral limits from 0 to b:[(4/3)b3 - 12b2 + 36b] - [(4/3)(0)3 - 12(0)2 + 36(0)] = 36(4/3)b3 - 12b2 + 36b = 36To simplify, divide the entire equation by 4:(1/3)b3 - 3b2 + 9b = 9Multiply by 3 to clear the fraction:b3 - 9b2 + 27b = 27Rearrange the equation to set it to zero:b3 - 9b2 + 27b - 27 = 0Recognize this as the binomial expansion of (b-3)3:(b-3)3 = 0Taking the cube root of both sides:b - 3 = 0b = 3Thus, the value of b is 3. This matches option B.