Final answer:
To compute the definite integral ∫[0,2] (x²/3 + 8) dx using the Riemann sum approach, we divide the interval [0,2] into n subintervals and evaluate the sum at the right endpoints, which converges to the integral as n approaches infinity.
Step-by-step explanation:
The student is asking how to compute the definite integral ∫[0,2](x²/3 + 8) dx using a limit of Riemann sums as the number of subintervals goes to infinity. To solve the problem, we'll set up the Riemann sum for the given function f(x) = (x²/3) + 8 on the closed interval [0,2]. We'll use the right endpoints for each subinterval to evaluate the sum.
The limit of the Riemann sum as n approaches infinity is the definition of the definite integral: limn→∞ ∑i=1f(xi)Δx.
To compute this, we divide the interval [0,2] into n equally spaced subintervals, each of width Δx = (2-0)/n = 2/n. The right endpoint of the i-th subinterval is xi = 2i/n. Thus, our Riemann sum looks like ∑i=1(xi²/3 + 8) * (2/n).
Substitute xi = 2i/n into the function to get ∑i=1(((2i/n)²/3) + 8)*(2/n), and simplify to obtain the sum Sn = (∑i=1) ((8i2)/(3n3) + (16i)/n2). As n approaches infinity, this sum converges to the definite integral, which can be evaluated further to find the exact value by using antiderivatives and evaluating at the endpoints x=0 and x=2.