How to determine which of the following regions has an area equal to the given limit without evaluating the limit ?

Question

William Abdalla · Accepted Answer

This is the limit of a Riemann Sum for the function #f(x)=x^10# on the interval #[5,7]#

A Riemann sum has the form

#sum_(i=1)^n f(x_i)Deltax#

Where #Deltax = (b-a)/n#. We can interpret this expression as having #Deltax = 2/n#, so #b-a=2#.

If we partition an interval #[a,b]# into #n# subintervals of equal length, then we can find the left endpoints of the subintervals using #a+iDeltax# to find the right endpoint of subinterval #i#. In this expression we can interpret #5+(2i)/n# as this calculation.

Finally, we see the the expression has #(x_i)^10#, so we must have #f(x)=x^10#

Therefore, it represents the area under the curve #y=x^10# and above the #x#-axis between #x=5# and #x=7#.

Elijah Abram · Answer

undefined To determine which region has an area equal to a given limit without evaluating the limit, you can compare the functions representing the regions and analyze their behavior. Look for patterns, symmetries, or specific characteristics of the functions that can help determine the area. Additionally, you can use mathematical techniques such as integration or geometric properties to calculate the area of the regions.

Adam Adkins · Answer

undefined To determine which of the given regions has an area equal to the given limit without evaluating the limit directly, you can use the concept of Riemann sums and integrals. First, express the limit as an integral, then compare it to the integrals representing the areas of the given regions. Evaluate the integrals for each region without substituting the limit value and compare the results to identify the region whose area matches the given limit.