# What is lower Riemann sum?

See below

The lower Riemann sum for the integral

involves

(some authors call the result of the summation the lower Riemann sum, the limit being called the lower Riemann integral)

For the integral to exist, both the lower and the upper Riemann sum must exist and be equal - their common value is the Riemann integral.

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A lower Riemann sum is a method for approximating the area under a curve using rectangles. It involves dividing the interval over which the function is being integrated into subintervals and then constructing rectangles based on the minimum function value within each subinterval. The height of each rectangle is determined by the minimum function value within the corresponding subinterval. The sum of the areas of these rectangles gives an approximation of the total area under the curve. As the number of subintervals increases, the lower Riemann sum approaches the actual area under the curve (the integral) if the function is Riemann integrable.

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The lower Riemann sum is a way to approximate the area under a curve using rectangles. It is obtained by partitioning the interval over which the function is defined into subintervals, then constructing rectangles whose heights are determined by the minimum value of the function within each subinterval. The sum of the areas of these rectangles gives an approximation to the area under the curve. As the number of subintervals increases, this approximation becomes more accurate, converging to the actual area under the curve.

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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.

- How do you find the area using the trapezoidal approximation method, given #(x^2-x)dx#, on the interval [0,2] with n=4?
- How do you use the Trapezoidal Rule with step size n=4 to estimate #int t^3 +t) dx# with [0,2]?
- Use Riemann sums to evaluate? : #int_0^3 \ x^2-3x+2 \ dx #
- How do you use the trapezoidal rule to approximate integral of #e^-3x^2 dx# between [0,1]?
- How do you use a Riemann Sum with n = 4 to estimate #ln3 = int (1/x)# from 1 to 3 using the right endpoints and then the midpoints?

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