How do you find Find the Riemann sum that approximates the integral #int_0^9sqrt(1+x^2)dx# using left endpoints with #n=6#?

To find the Riemann sum that approximates the integral $\int_{0}^{9} \sqrt{1 + x^2} \, dx$ using left endpoints with $n = 6$, you need to partition the interval $[0, 9]$ into $n$ subintervals of equal width and then evaluate the function at the left endpoint of each subinterval.

The width of each subinterval, $\Delta x$, is given by:

$\Delta x = \frac{9 - 0}{6} = \frac{9}{6} = \frac{3}{2}$

So, the six subintervals are:

$[0, \frac{3}{2}], [\frac{3}{2}, 3], [3, \frac{9}{2}], [\frac{9}{2}, 6], [6, \frac{15}{2}], [\frac{15}{2}, 9]$

Now, evaluate the function at the left endpoint of each subinterval:

$f(0) = \sqrt{1 + (0)^2} = \sqrt{1} = 1$
$f\left(\frac{3}{2}\right) = \sqrt{1 + \left(\frac{3}{2}\right)^2} = \sqrt{\frac{13}{4}}$
$f(3) = \sqrt{1 + 3^2} = \sqrt{10}$
$f\left(\frac{9}{2}\right) = \sqrt{1 + \left(\frac{9}{2}\right)^2} = \sqrt{\frac{85}{4}}$
$f(6) = \sqrt{1 + 6^2} = \sqrt{37}$
$f\left(\frac{15}{2}\right) = \sqrt{1 + \left(\frac{15}{2}\right)^2} = \sqrt{\frac{221}{4}}$

Finally, compute the Riemann sum:

$\text{Riemann Sum} = \Delta x \left( f(0) + f\left(\frac{3}{2}\right) + f(3) + f\left(\frac{9}{2}\right) + f(6) + f\left(\frac{15}{2}\right) \right)$