Integral problem...I need help!?

Question

I'm working for a problem and I need some help...

Consider the definite integral: (2X+1)dx with b=3,a=1

I used the fundamental theorem and got the answer 10

but when I used riemann sum I got 12.

Why is that?

Answer 1

Which x_i* did you choose for the Riemann sum?

Since the function is increasing, if you chose the right endpoint of each subinterval as x_i*, you should expect the Riemann sum to be too large (see this picture: http://www.uwm.edu/Dept/Math/Resources/Calculus/Key/4b.gif ). If you chose the left endpoint, you'd expect it to be too small (see this picture: http://www.uwm.edu/Dept/Math/Resources/Calculus/Key/4a.gif ). Also how many subintervals did you choose? The smaller the subinterval, the more accurate the answer you get.

The Riemann sum is the approximation whose limit becomes the definite integral...so most times the Riemann sum will be larger or smaller, right?

Disclaimer: I haven't had calc since 1991.

Answer 2

a riemann sum is ONLY an APPROXIMATION,
so depending on how many subdivisions you use, you get a better (more subdivisions) or worse (fewer subdivision) approximation to the value.
now, the fundamental theorem gives the exact value of the integral.

interal from 1 to 3 2x+ 1 dx
= x^2 + x from 1 to 3
= 9 +3 - ( 1+1)
=10 .

Answer 3

the riemann sum is an approximate answer, and it depends on how small the intervals are, the closer your answer becomes. the fundamental theorem will give you an exact answer

Answer 4

the answer is 10 but I am too bored to repeat my university math books to answer this.I coudl give you a guess though...Probably you are not doing smth right with the Riemann sum.As far as I remember it was quite complicated to do Riemann summs.Our prof would give examples with even simpler functions than that such as x^3 and would still take him a blackboard to write the solution...
So, since the solution is 10 for sure, and since the Riemann sum should give you the right solution, there is where your mistake is.

Answer 5

In the Reiman sum, your taking the mid point, or left side or right side of each rectangle.
The area that you leave out and the area that you include are different.
In fundamental theorum, you take the limit of the Rieman Sum as n (number of rectangles) approaches infinity - in other words as the width of the rectangle reaches a straight line.