Determine a region whose area is equal to the given limit. Do not evaluate the limit. limit as n approaches infinity of&nbsp;∑i=1n(2/n)((5+(2i)/n)10)

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Determine a region whose area is equal to the given limit. Do not evaluate the limit. limit as n approaches infinity of&amp;nbsp;∑i=1n(2/n)((5+(2i)/n)10)

Jazz Frenia · Accepted Answer

We are given the limit:limn→∞∑i=1n(2n)(5+2in)10We need to determine a region whose area is equal to this limit.Let&#039;s rewrite the limit as an integral:limn→∞∑i=1n(2n)(5+2in)10=limn→∞(2n)∑i=1n(5+2in)10n2⇒limn→∞∑i=1n(5+2in)101n=∫02(5+2x2)10dxHere, we used the fact that 2n is the width of each rectangle in the Riemann sum and 5+2in is the height of each rectangle. As n approaches infinity, the width of each rectangle becomes infinitesimally small and the sum approaches the integral.Now, we can evaluate the integral:∫02(5+2x2)10dx=[111(5+2x2)11]02=111(5+2)11−111(5+0)11=111(711−511)Therefore, the region whose area is equal to the given limit is the area under the curve y=(5+2x2)10 from x=0 to x=2, which has an area of 111(711−511).

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