Question

Riemann Sums. I am not sure if I am calculating this correctly.

Answer 1

Using the left-side Riemann Sum.

Answer 2

What have you tried?

Answer 3

I get 0. But I don't think it is right.

Answer 4

Hi.
Do you have a certain number N that they want you to calculate for?
N=1... N=2... N=...?

Answer 5

Got it... 3 subintervals.

Answer 6

So 3 subintervals. X goes from -3 to 15... That is a total displacement of 18.
18 divided by 3 subintervals means you need a displacement of 6 for each subinterval from the left side.

We know that Riemann Sums use the origin as the base of a shape, be it triangle, square, elliptical, etc.

Go from -3 to 3... Go from 3 to 9... Go from 9 to 15...
From -3 to 3, we see it has some weird elliptical, plus another one of them of the same exact length, but positive....Do you see it?
The elliptical is from -3 to 0, and the elliptical is from 0 to 3.
Let's treat it as two triangles that go from negative, and then positive.

-3 -> 3.
Triangle:-3 to 0 would be 3 times 1 times 1/2, which would give you -3/2.
2Triangle:0 to 3 would be 3 times 1 times 1/2, which would give you 3/2.

Let's do 3->9...

Answer 7

Er... you said LHS. Okay, forget 2Triangle. Make it a square instead. That would be a simple -3 times 1, which would give you an area of -3.

So -3->3.
Triangle: -3 to 0 would be 3 times 1 times 1/2, which would give you -3/2.
Square: 0 to 3 would be -3 times 1, which would give you an area of -3.

Let's do 3-> 9.

Answer 8

and what about the triangle above the square? above the x-axis?

Answer 9

You'll understand in my explanation for 3->9, one sec.

Answer 10

Do the LHS from 3 to 9.
You see a simple rectangle. Everything above it? Forget it! Doesn't exist right now. You're doing a LEFT hand sum. Meaning you start from the leftmost point, the lowest point, and work your way to the desired end-x-value. So at +3, your leftmost point is at a height of 1, and you draw your rectangle all the way to the end-x-value of +9. That would give you a rectangle with a dimension of a 6x1.

So 3->9
Rectangle has a height of 1 and a length of 6.
Your area is positive 6.

Let's do the ending, 9 to 15...

Answer 11

so the bottom doesn't matter?

Answer 12

and from 9 to 15 would be a positive 6?

Answer 13

From 9 to 15, we see that there is a positive section of the graph (looks like an elliptical) and the negative, opposite of that. They cancel out. Your total area is zero for that, since it goes from positive to negative at the exact middle.

Here...


If I were to tell you to do an LHS from zero to four, you will have to start at the leftmost point at zero, and reach all the way to four, and then drop to the origin.

Answer 14

The left hand sum of this would be a rectangle from positive Y to 4, dimensions 4xY... Lets say the height is 1, your LHS would be 4x1 = 4...
Your right hand sum would the rightmost X value, which would be 4, going to zero... That would be dimensions 4xY. Lets say Y is negative one. Your area would be 4x-1 = -4.
Your actual integral would be the area most accurately defining this figure... You would be doing the area from zero to two, then the area of two to six, then the area of six to eight, then the area of eight to ten...

Riemann sums are very inaccurate, especially when the interval number is very small... The more intervals, the more accurate. So your sqrt98 would come off to be a strange number. You'd think it would be somewhere between 9 and 10, right? Well, with three approximations and an LHS, you'd be far under that value... Hence, LHS means underestimate of an answer. RHS means over... MHS (not sure if you've done this) is the most accurate, but nowhere near as precise as an actual integral.

Answer 15

so the sum is 0 after all that?

Answer 16

From sqrt(98)? Nah.
-3 to 3, sum is (-3) + (-3/2). = -9/2.
From 3 to 9, sum is (1)*(6) = 6.
From 9 to 15, sum is (+area)+(-area)=0

Ans. 3/2.

What would make it more of a realistic answer? Use more intervals! If you used six subintervals, you'd have an answer more realistic. If you used 500000000000 sub intervals (you'll learn how to do this on a calculator. I can tell you the equation if you'd like to know) then you'd probably have a very realistic answer.

Answer 17

Not sure if you're still there, but the answer is six... I forgot to do the LHS from the right side, of 9->15. This area is the same as from -3->6.
Therefore your answer is 6! Not 3/2.


-3 to 3, sum is (-3) + (-3/2). = -9/2.
From 3 to 9, sum is (1)*(6) = 6.
From 9 to 15, sum is (3/2)+(3) = 9/2.

Ans. 6.

This is still an underestimate.