document.write( "Question 426974: suppose you estimate the area under a curve using inscribed rectangles and then estimate the area using circumscribed rectangles. how will the two estimates for the area compare? \n" ); document.write( "

Algebra.Com's Answer #296945 by robertb(5830) $\"\"$ $\"About$

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Let $\"S%5BL%5D+\"$ = sum of the areas of the inscribed rectangles (also called the lower Riemann sum), and \r
\n" ); document.write( "\n" ); document.write( " $\"S%5BU%5D\"$ = sum of the areas of the circumscribed rectangles (also called the upper Riemann sum).\r
\n" ); document.write( "\n" ); document.write( "Then if the curve $\"y+=+f%28x%29+%3E=+0\"$ over the interval [a,b], (and using the same partitioning of the interval) then $\"S%5BU%5D+%3E=+S%5BL%5D\"$ . As the partitioning gets finer (i.e., more subintervals), the two sums should converge to a common value which is the definite integral of f(x) over [a,b] if y = f(x) is integrable over [a,b]. (Note that if $\"y+=+f%28x%29+%3C=+0\"$ then the relationship between the two Riemann sums is also reversed.) \n" ); document.write( "

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