Approximate the area under the graph of 𝑓(𝑥) = x^2 + 1 from x = –1 to x = 2. Answer: a.) Sketch the graph and the rectangles then estimate the area by using the left endpoints of six subintervals.

Answer 1

I've recorded an explanation of the idea behind the answer here. If you want to know more about Riemann sums and how to conceptualize them, please let me know!

Answer 2

x^3 + x evaluated -1 to 2 = 10/3 +2/3 + 1/3 +1 = 6

6 is the exact area. doing the intervals from the left side should be slightly less than 6, an underestimate

6 intervals means each with length 1/2

left ends mean height of 2, 1/4+1, 1, 1/4 + 1, 1+1, 9/4 + 1,

or 2, 5/4, 1, 5/4, 2, 13/4,

each times 1/2 = (1/2)(2 +1+2 + 10/4 +13/4) = (1/2)(5+ 23/4) = 2.5 + 23/8 = 2.5 + 2.875 = 5.375

left side underestimates, right side overestimates.

left side <6, right side>6