Problem 7 - Calculus Early Transcendentals

Question

Does a right Riemann sum underestimate or overestimate the area of the region under the graph of a function that is positive and decreasing on an interval \([a, b] ?\) Explain.

Step-by-Step Solution

Answer

Answer: Underestimate

Step 1: Define a Right Riemann Sum

A right Riemann sum is an approximation of the area under the curve of a function by dividing the interval \([a, b]\) into smaller subintervals and forming rectangles. The height of each rectangle is determined by the function's value at the right endpoint of the corresponding subinterval, and the width of each rectangle is the length of the subinterval. The sum of the areas of these rectangles gives the right Riemann sum.

Step 2: Characteristics of a Positive and Decreasing Function

A positive and decreasing function is a function that is always above the \(x\)-axis (positive) and whose values decrease as we move from left to right along the interval \([a, b]\).

Step 3: Relationship Between Right Riemann Sum and Function

For a positive and decreasing function, the value of the function decreases as we move along the interval from \(a\) to \(b\). Therefore, the height of the rectangles used in the right Riemann sum will decrease as we move from left to right.

Step 4: Compare Rectangles with Actual Function

When we compare the rectangles of the right Riemann sum to the actual shape of the function, we can see that for each subinterval, the rectangle's height is the function's value at the right endpoint, so there is no extra area added above the curve. On the other hand, there might be some area below the curve that is not covered by the rectangles, as the curve has already decreased to a smaller value than the height of the rectangle on the left side of the subinterval.

Step 5: Conclusion: Underestimate or Overestimate?

Since there is no extra area above the curve added by the rectangles and there might be some area below the curve not covered by the rectangles, we can conclude that a right Riemann sum underestimates the area under the graph of a function that is positive and decreasing on an interval \([a, b]\).