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Areas of regions under curves can be approximated by evaluating Riemann sums, which give the area of a collection of rectangles. In this project, we investigate special cases in which is it possible to compute areas of regions exactly using Riemann sums. Rather than just taking large values of \( n \) (the number of rectangles), we actually evaluate the limit as \( n \rightarrow \infty \). As an example, consider the definite integral \( \int_{2}^{4}(2 x+3) d x \), which is the area of the region bounded by the graph of \( f(x)=2 x+3 \), the \( x \)-axis, and the lines \( x=2 \) and \( x=4 \). 1. Partition the interval \( [2,4] \) into \( n \) subintervals. Each subinterval will serve as the width of each of the rectangles used for the Riemann sum approximating the area of the region. If each width is taken to be of equal measure, then let the widths be \( \Delta x=\frac{4-2}{n}=\frac{2}{n} \).

          Areas of regions under curves can be approximated by evaluating Riemann sums, which give the area of a collection of rectangles. In this project, we investigate special cases in which is it possible to compute areas of regions exactly using Riemann sums. Rather than just taking large values of \( n \) (the number of rectangles), we actually evaluate the limit as \( n \rightarrow \infty \).
As an example, consider the definite integral \( \int_{2}^{4}(2 x+3) d x \), which is the area of the region bounded by the graph of \( f(x)=2 x+3 \), the \( x \)-axis, and the lines \( x=2 \) and \( x=4 \).
1. Partition the interval \( [2,4] \) into \( n \) subintervals. Each subinterval will serve as the width of each of the rectangles used for the Riemann sum approximating the area of the region. If each width is taken to be of equal measure, then let the widths be \( \Delta x=\frac{4-2}{n}=\frac{2}{n} \).

Areas of regions under curves can be approximated by evaluating Riemann sums, which give the area of a collection of rectangles. In this project, we investigate special cases in which is it possible to compute areas of regions exactly using Riemann sums. Rather than just taking large values of n (the number of rectangles), we actually evaluate the limit as n →∞.
As an example, consider the definite integral ∫2^4(2 x+3) d x, which is the area of the region bounded by the graph of f(x)=2 x+3, the x-axis, and the lines x=2 and x=4.
1. Partition the interval [2,4] into n subintervals. Each subinterval will serve as the width of each of the rectangles used for the Riemann sum approximating the area of the region. If each width is taken to be of equal measure, then let the widths be Δ x=(4-2)/(n)=(2)/(n).

Added by Darnell C.

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