Riemann Sum Calculator for a Table

For the given table of values, the calculator will approximate the definite integral using the Riemann sum and sample points of your choice: left endpoints, right endpoints, midpoints, and trapezoids.

Related calculator: Riemann Sum Calculator for a Function

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Your Input

Approximate the integral $$$\int\limits_{0}^{8} f{\left(x \right)}\, dx$$$ with the left Riemann sum using the table below:

$$$x$$$$$$0$$$$$$2$$$$$$4$$$$$$6$$$$$$8$$$
$$$f{\left(x \right)}$$$$$$1$$$$$$-2$$$$$$5$$$$$$0$$$$$$7$$$

Solution

The left Riemann sum approximates the integral using left endpoints: $$$\int\limits_{a}^{b} f{\left(x \right)}\, dx\approx \sum_{i=1}^{n - 1} \left(x_{i+1} - x_{i}\right) f{\left(x_{i} \right)}$$$, where $$$n$$$ is the number of points.

Therefore, $$$\int\limits_{0}^{8} f{\left(x \right)}\, dx\approx \left(2 - 0\right) 1 + \left(4 - 2\right) \left(-2\right) + \left(6 - 4\right) 5 + \left(8 - 6\right) 0 = 8$$$.

Answer

$$$\int\limits_{0}^{8} f{\left(x \right)}\, dx\approx 8$$$A