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Every man is a mammal, but not every mammal is a man At 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 \( \Delta x=\frac{4-2}{n}=\frac{2}{n} \). 2. Let's look at right Riemann sums. The right endpoint \( x_{k}^{} \) of the general \( k \) th subinterval is \( x_{k}^{}=2+k \cdot \Delta x \), where \( k=1,2, \ldots, n \). That gives us \( f\left(x_{k}^{}\right) \) as the heights of the approximating rectangles used for the Riemann sum. Evaluate \( f\left(x_{k}^{}\right) \) to find an expression of the height for each rectangle only in terms of the variables \( n \) and \( k \). 3. If \( \Delta x \) is the width and \( f\left(x_{k}^{}\right) \) is the height of the general approximating rectangle of the Riemann sum, then \( f\left(x_{k}^{}\right) \cdot \Delta x \) is the area of that rectangle. Find an expression of the area \( f\left(x_{k}^{}\right) \cdot \Delta x \) of each rectangle only in terms of \( n \) and \( k \). 4. Next, calculate the sum of \( n \) approximating rectangles of the Riemann sum \( R_{n}=\sum_{k=1}^{n} f\left(x_{k}^{}\right) \cdot \Delta x \) and write your answer only in terms of \( n \). 5. Finally, evaluate the limit of the finite Riemann sum, \( \lim _{n \rightarrow \infty} R_{\mathrm{n}} \). This is the exact area of the region bounded by the graph of \( f(x)=2 x+3 \), the \( x \)-axis, and the lines \( x=2 \) and \( x=4 \).

          Every man is a mammal, but not every mammal is a man At
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 \( \Delta x=\frac{4-2}{n}=\frac{2}{n} \).
2. Let's look at right Riemann sums. The right endpoint \( x_{k}^{*} \) of the general \( k \) th subinterval is \( x_{k}^{*}=2+k \cdot \Delta x \), where \( k=1,2, \ldots, n \). That gives us \( f\left(x_{k}^{*}\right) \) as the heights of the approximating rectangles used for the Riemann sum. Evaluate \( f\left(x_{k}^{*}\right) \) to find an expression of the height for each rectangle only in terms of the variables \( n \) and \( k \).
3. If \( \Delta x \) is the width and \( f\left(x_{k}^{*}\right) \) is the height of the general approximating rectangle of the Riemann sum, then \( f\left(x_{k}^{*}\right) \cdot \Delta x \) is the area of that rectangle. Find an expression of the area \( f\left(x_{k}^{*}\right) \cdot \Delta x \) of each rectangle only in terms of \( n \) and \( k \).
4. Next, calculate the sum of \( n \) approximating rectangles of the Riemann sum \( R_{n}=\sum_{k=1}^{n} f\left(x_{k}^{*}\right) \cdot \Delta x \) and write your answer only in terms of \( n \).
5. Finally, evaluate the limit of the finite Riemann sum, \( \lim _{n \rightarrow \infty} R_{\mathrm{n}} \). This is the exact area of the region bounded by the graph of \( f(x)=2 x+3 \), the \( x \)-axis, and the lines \( x=2 \) and \( x=4 \).

Every man is a mammal, but not every mammal is a man At
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).
2. Let's look at right Riemann sums. The right endpoint xk^* of the general k th subinterval is xk^*=2+k ·Δ x, where k=1,2, …, n. That gives us f(xk^*) as the heights of the approximating rectangles used for the Riemann sum. Evaluate f(xk^*) to find an expression of the height for each rectangle only in terms of the variables n and k.
3. If Δ x is the width and f(xk^*) is the height of the general approximating rectangle of the Riemann sum, then f(xk^*) ·Δ x is the area of that rectangle. Find an expression of the area f(xk^*) ·Δ x of each rectangle only in terms of n and k.
4. Next, calculate the sum of n approximating rectangles of the Riemann sum Rn=∑k=1^n f(xk^*) ·Δ x and write your answer only in terms of n.
5. Finally, evaluate the limit of the finite Riemann sum, limn →∞ Rn. This is the exact 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.

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