Define g(x) = 2x for x ∈ [0, 1]. Use the definition of the Riemann integral to show that g ∈ R[0, 1]and thatZ 10g = 1

To show that g ∈ R[0, 1] and ∫ from 0 to 1 g = 1, we need to use the definition of the Riemann integral.

Step 1: Define the function We have g(x) = 2x for x ∈ [0, 1].

Step 2: Partition the interval [0, 1] We partition the interval [0, 1] into n subintervals, each of width Δx = 1/n. The kth subinterval is [x_k-1, x_k], where x_k = k/n for k = 1, 2, ..., n.

Step 3: Choose sample points In each subinterval [x_k-1, x_k], we choose a sample point c_k. In this case, we can simply choose c_k = x_k.

Step 4: Form the Riemann sum The Riemann sum of g with respect to this partition and these sample points is R = Σ from k=1 to n g(c_k) Δx = Σ from k=1 to n 2c_k * (1/n) = 2/n * Σ from k=1 to n k/n = 2/n^2 * Σ from k=1 to n k.

Step 5: Evaluate the Riemann sum The sum of the first n integers is n(n+1)/2, so R = 2/n^2 * n(n+1)/2 = (n+1)/n = 1 + 1/n.

Step 6: Take the limit as n goes to infinity As n goes to infinity, 1/n goes to 0, so the limit of R as n goes to infinity is 1.

Therefore, by the definition of the Riemann integral, g is Riemann integrable on [0, 1] and ∫ from 0 to 1 g = 1.