What is midpoint Riemann sum?

Answer 1

I will assume that you know the general idea for a Riemann sum.

It is probably simplest to show an example:

For the interval: #[1,3]# and for #n=4#
we find #Delta x# as always for Riemann sums:
#Delta x = (b-a)/n = (3-1)/4 = 1/2#

Now the endpoints of the subintervals are:

#1, 3/2, 2, 5/2, 2#

The first four are left endpoint and the last four are right endpoints of subintervals.

The left Riemann sum uses the left endpoints to find the height of the rectangle. (And the right sum . . . )

The midpoint sum uses the midpoints of the subintervals:

#[1, 3/2]# #[3/2,2]# #[2,5/2]# #[5/2, 3]#

The midpoint of an interval is the average (mean) of the endpoints:

#m_1 = 1/2(1+3/2) = 5/4#
#m_2 = 1/2(3/2 + 2) = 7/4#
#m_3 = 1/2 (2+5/2) = 9/4#
#m_4 = 1/2 (5/2+3) = 11/4#
Now, whatever the function #f#, we get the sum:
#Delta x (f(m_1) + f(m_2) + f(m_3)+f(m_4))#
#= 1/2(f(5/4)+f(7/4)+f(9/4)+f(11/4))#
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Answer 2

Midpoint Riemann sum is a method for approximating the area under a curve using rectangles. In this method, the width of each rectangle is determined by the width of the subintervals into which the interval of integration is divided. The height of each rectangle is determined by the function evaluated at the midpoint of each subinterval. These rectangles are then summed to estimate the total area under the curve. Mathematically, the midpoint Riemann sum is expressed as the sum of the products of the function values at the midpoints of the subintervals and the width of each subinterval. This approximation becomes more accurate as the number of subintervals increases.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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