Final answer:
To evaluate the integral of x²(4x+3) from 2 to 3, compute the limit of a Riemann sum, resulting in the integral being equal to 51.
Step-by-step explanation:
The question asks to evaluate the integral of x²(4x+3) from 2 to 3 by computing the limit of a Riemann sum. To solve this, we divide the interval [2,3] into n subintervals of equal width Δx. The Riemann sum is then the sum of the areas of rectangles with height x²(4x+3) and width Δx, evaluated at points within each subinterval. As n approaches infinity, the Riemann sum approaches the exact area under the curve, which is the value of the integral. Specifically, the limit of the Riemann sum as n approaches infinity gives us the definite integral:
∫₂³ x²(4x+3) dx = ∫₂³ (4x³ + 3x²) dx = [rac{4}{4}x⁴ + rac{3}{3}x³]⁴₂ = (64 + 27) - (32 + 8) = 91 - 40 = 51.
Therefore, the value of the integral is 51.