# Riemann Sum Approximation (confused…)

I'm having trouble determining when a left-hand approximation is an over/under estimate for a given function.

For example, for the graph above $f(x)$=$ln(x)$ and the function is increasing for all $x$ > 0.Since the function continues to increase, doe that mean the left-hand approximation would be an underestimate? $$\\$$ Update:

• Yes. Try drawing the rectangles for a Riemann sum approximation. Since they are 'anchored' at the left side, the curve lies above the rectangles, and therefore its area exceeds the areas of the rectangles. – Nick Peterson Jul 11 '17 at 5:35
• @NickPeterson so would that mean the right-hand approximation would be an overestimate? – AmR Jul 11 '17 at 5:35
• Yep. Again: draw the rectangles. – Nick Peterson Jul 11 '17 at 5:36
• @NickPeterson is the updated image in my question correct for the left-hand? – AmR Jul 11 '17 at 5:42
• Yep! That's the image I was suggesting. – Nick Peterson Jul 11 '17 at 5:53

When the function is always decreasing, that means the right-hand sum will be an underestimate and the left-hand sum will be an overestimate. $$\\$$
For the function $f$($x$)=$ln$($x$), it is always increasing.