Final answer:
To find the Riemann sum for the function f(x) = 7sin(x) with six terms and right endpoints as the sample points, divide the interval [0, 3π/2] into six subintervals of equal length. Calculate the right endpoints and evaluate the function at each right endpoint, and then multiply by the width of the subinterval. For the Riemann sum with midpoints as the sample points, calculate the midpoints and evaluate the function at each midpoint, multiply by the width of the subinterval, and add up the products.
Step-by-step explanation:
To find the Riemann sum for the function f(x) = 7sin(x) with six terms and right endpoints as the sample points, we need to divide the interval [0, 3π/2] into six subintervals of equal length. The width of each subinterval would be Δx = (3π/2 - 0)/6 = π/4. Then, we can calculate the right endpoint of each subinterval by adding the width to the left endpoint: x1 = 0 + π/4, x2 = π/4 + π/4, x3 = π/2 + π/4, and so on up to x6 = 3π/2 - π/4. Next, for each subinterval, we evaluate the function at the corresponding right endpoint and multiply it by the width of the subinterval. Finally, we add up all these products to get the Riemann sum.
For the second part, where we take midpoints as the sample points, the process is similar. We divide the interval into six subintervals of width π/4. Then, we calculate the midpoint of each subinterval by taking the average of the left and right endpoint: m1 = (0 + π/4)/2, m2 = (π/4 + π/2)/2, and so on up to m6 = (3π/2 - π/4 + 3π/2)/2. Finally, we evaluate the function at each midpoint and multiply it by the width of the subinterval and add up all these products to obtain the Riemann sum with midpoints as the sample points.