Left Endpoint Approximation Calculator for a Table

For the given table of values, the calculator will approximate the integral using the left endpoints (the left Riemann sum), with steps shown.

Number of points:

Points:

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

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Your Input

Approximate the integral $$$\int\limits_{-3}^{5} f{\left(x \right)}\, dx$$$ with the left endpoint approximation using the table below:

$$$x$$$	$$$-3$$$	$$$-2$$$	$$$0$$$	$$$3$$$	$$$5$$$
$$$f{\left(x \right)}$$$	$$$-2$$$	$$$3$$$	$$$-1$$$	$$$2$$$	$$$5$$$

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, $$$\int\limits_{-3}^{5} f{\left(x \right)}\, dx\approx \left(-2 - \left(-3\right)\right) \left(-2\right) + \left(0 - \left(-2\right)\right) 3 + \left(3 - 0\right) \left(-1\right) + \left(5 - 3\right) 2 = 5.$$$

Answer

$$$\int\limits_{-3}^{5} f{\left(x \right)}\, dx\approx 5$$$A