009B Sample Midterm 2, Problem 1

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Consider the region bounded by and the -axis.

a) Use four rectangles and a Riemann sum to approximate the area of the region . Sketch the region and the rectangles and indicate whether your rectangles overestimate or underestimate the area of .
b) Find an expression for the area of the region as a limit. Do not evaluate the limit.

Link to Riemann sums page



Step 1:  
Let . Since our interval is and we are using 4 rectangles, each rectangle has width 1. Since the problem doesn't specify, we can choose either right- or left-endpoints. Choosing left-endpoints, the Riemann sum is
Step 2:  
Thus, the left-endpoint Riemann sum is
The left-endpoint Riemann sum overestimates the area of .


Step 1:  
Let be the number of rectangles used in the left-endpoint Riemann sum for .
The width of each rectangle is .
Step 2:  
So, the left-endpoint Riemann sum is
Now, we let go to infinity to get a limit.
So, the area of is equal to .
Final Answer:  
(a) The left-endpoint Riemann sum is , which overestimates the area of .
(b) Using left-endpoint Riemann sums:

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