# 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#?

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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) ]

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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.

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