Riemann Sum Calculator for a Table

Approximate an integral (given by a table of values) using the Riemann sum step by step

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

Related calculator: Riemann Sum Calculator for a Function

Number of points:

Points:

$$$x$$$
$$$f{\left(x \right)}$$$

Type:

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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