Compute the average of the RRAM and LRAM approximations to estimate the area between the graph of the function
and the x-axis over the given interval using the indicated number of subintervals. (The function is non-negative on the
given interval).
-$f ( x ) = 2 x ^ { 3 } + 4 ; [ 0,3 ] ; 3$ subintervals
A) 60
B) 54
C) 57
D) 63

Question

Quizplus · Accepted Answer

To find the average of the RRAM and LRAM approximations, we first calculate each approximation separately and then average them. The interval [0,3] is divided into 3 subintervals: [0,1], [1,2], and [2,3].For LRAM (Left Riemann Sum), we use the left endpoints of each subinterval to calculate the function's value:- For [0,1], $f(0) = 2(0)^3 + 4 = 4$- For [1,2], $f(1) = 2(1)^3 + 4 = 6$- For [2,3], $f(2) = 2(2)^3 + 4 = 20$LRAM = $4*1 + 6*1 + 20*1 = 30$For RRAM (Right Riemann Sum), we use the right endpoints of each subinterval:- For [0,1], $f(1) = 6$- For [1,2], $f(2) = 20$- For [2,3], $f(3) = 2(3)^3 + 4 = 62$RRAM = $6*1 + 20*1 + 62*1 = 88$Average = $\frac{LRAM + RRAM}{2} = \frac{30 + 88}{2} = \frac{118}{2} = 59$However, there seems to be a calculation mistake in the final step. Correctly calculating the average of the given values:LRAM = $4 + 6 + 20 = 30$RRAM = $6 + 20 + 62 = 88$Average = $\frac{30 + 88}{2} = \frac{118}{2} = 59$Given the options, it appears there was a mistake in my explanation regarding the final average calculation. The correct process was described, but the final answer should align with the options provided, indicating a need to reevaluate the calculations or the interpretation of the options. The correct approach involves calculating the function values at the specified points, summing them according to the LRAM and RRAM methods, and then averaging those sums. The discrepancy suggests a reevaluation or clarification of the calculation steps or the options given.

Compute the Average of the RRAM and LRAM Approximations to Estimate