Estimations using left endpoint rectangles

Select Riemann... on the Calculus Menu. Then select "Left endpoints with graph" ( found in the middle column ), enter f as the expression, x as the variable, from 0 to 2 with 4 rectangles.

> with(math191):

> gLsum(f,x=0..2,4);

Repeat what you just did but now use 8 left endpoint rectangles.

> gLsum(f,x=0..2,8);

Now try doubling the number of rectangles to 16.

> gLsum(f,x=0..2,16);

The Riemann... button also allows us to get estimates without making the graph. This will be quicker when the number of rectangles is large. Select "Left endpoints"( in the first column ) using the Riemann... button and use 16 rectangles again.

> Lsum(f,x=0..2,16);

Note that we have the result in rational form. Execute the following loop to get left endpoint approximations using 1,2,4,8,16,32,64,...,512 rectangles. The evalf that appears in the code will convert the fraction into a decimal.

> for i from 1 to 10 do
k:=2^(i-1):
evalf(Lsum(f,x=0..2,k)):
od;

What do you think these will approach?