Question

Evaluate the Riemann sum for (x) = x3 − 6x, for 0 ≤ x ≤ 3 with six subintervals, taking the sample points, xi, to be the right endpoint of each interval. Give three decimal places in your answer.

Answer 1

divide 3-0 by 6
to figure out what the length of each sub-interval should be
and just take the right endpoint of each sub-interval...

Answer 2

then once you have the right endpoint of each sub-interval
you can find the area of your 6 rectangles and add them up together

Answer 3

So basically I have an interval every half?

Answer 4

yes example interval 1 would be [0,1/2]
example 2 interval 2 would be [1/2,1]
and so on...

the right endpoints are 1/2,1,...

Answer 5

each sub-interval has length 1/2

Answer 6

so what would be the 3rd sub-interval?

Answer 7

1.5 no?

Answer 8

do you mean [1,1.5]
because a number by itself is not an interval

Answer 9

and the right endpoint of that interval is 1.5

Answer 10

Yeah sorry.

Answer 11

[0,1/2]
[1/2,1]
[1,3/2]
[3/2,2]
[2,5/2]
[5/2,3]
these are your 6 subinterval

Answer 12

you will consider the y values for each right endpoint of each of the sub-intervals

Answer 13

base of each rectangle is 1/2
you have to evaluate:
\[\frac{1}{2}(f(\frac{1}{2})+f(1)+f(\frac{3}{2})+f(2)+f(\frac{5}{2})+f(3)]\]

Answer 14

I believed when you typed this "(x) = x3 − 6x"
you meant f(x)=x^3-6x

Answer 15

You are right. Let me evaluate that.

Answer 16

Am I supposed to get a negative number?

Answer 17

that is what I have

Answer 18

we could get a better idea of why that is by graphing the thingy on [0,3]

Answer 19

Ok, but it is equal to -3.94?

Answer 20

it is definitely a net area
that area below the x-axis must have been bigger then the area of above the x-axis
since we got a negative result

and yes -3.94 is what I have too