Please answer fully. Three results for finding the area under the curve y = 3x,
ID: 2829193 • Letter: P
Question
Please answer fully.
Three results for finding the area under the curve y = 3x, between x = 0 and x = 3, are shown below. The result found by dividing the interval into 10 subintervals and then adding up the areas of the inscribed rectangles is 12.15. The result found by dividing the interval into 10 subintervals and then adding up the areas of the circumscribed rectangles is 14.85. The exact result found by evaluating the antiderivative at the bounds is 13.5. Why is the mean of the sums of the inscribed rectangles and circumscribed rectangles equal to the exact value?Explanation / Answer
Because y = 3x is a linear function, as a result the average of the heights of inscribed and circumscribed rectangles is equal to the value of the function. Since both methods use 10 intervals, the width of all rectangles is the same ((3-0)/10 = 0.3), which results in this statement.