Final answer:
To estimate the area between the function f(x)=9 cos (x/6) and the x-axis using four rectangles, divide the interval (0, 3π) into four equal subintervals, calculate the function values at the left endpoints, multiply each by the width of the subintervals, sum the results, and round as necessary.
Step-by-step explanation:
To estimate the area between the graph of the function f(x)=9 cos (x/6 ) and the x-axis over the interval (0, 3π) using four rectangles, we will use the Left Riemann Sum method. The interval (0, 3π) needs to be divided into four equal subintervals, since we are using four rectangles for the estimation. Each subinterval will have a width of (3π - 0)/4 = 3π/4.
The left endpoints for these subintervals will be x-values at: 0, 3π/4, 3π/2, and 9π/4. We calculate the height of each rectangle by evaluating the function f(x) at these x-values, so we have f(0), f(3π/4), f(3π/2), and f(9π/4). These heights represent the function values at the left endpoints of each subinterval.
The area of each rectangle is given by the product of its height and its width, and the total area is the sum of the areas of the four rectangles. Finally, we sum up all the individual areas to obtain an estimate for the total area between the curve f(x) = 9 cos (x/6) and the x-axis from x=0 to x=3π. Remember to round any intermediate calculations to at least six decimal places, and to round your final answer to three decimal places for accuracy.