This problem tests the skill of translating a Riemann sum into its corresponding definite integral. The factor $\frac{\pi}{2n}$ represents \Delta x, indicating the total interval length is \pi/2. The argument $\frac{\pi i}{2n}$ maps to x_i = i \Delta x, from near 0 to \pi/2. The function is \sin(x), with limits from 0 to \pi/2. A tempting distractor is choice D, \int_${0}^{\pi/2}$\sin(2x),dx, which fails by incorrectly doubling the argument of sine. A general strategy for translating Riemann sums is to identify \Delta x = $\frac{b-a}{n}$, the form of x_i = a + i \Delta x, and the function f(x_i), so the sum approximates \int_$a^b$ f(x) dx as n approaches infinity.

This question assesses the skill of translating a Riemann sum into its corresponding definite integral. The Riemann sum is \(\sum_{n=1}^{8} s(6+0.75n)(0.75)\), with \(\Delta x = 0.75\) as the width. The points are 6 + 0.75n for n from 1 to 8, from 6.75 to 12, covering 8 terms. This right Riemann sum corresponds to \(\int_{6}^{12} s(x),dx\), total length 8 × 0.75 = 6, from 6 to 12. A tempting distractor like choice B, \(\int_{6.75}^{12.75} s(x),dx\), fails by using points as limits and adding extra width. A general strategy for translating Riemann sums is to identify \(\Delta x\), count the terms to find the total length, and determine the limits by aligning the starting point with the overall interval.

This question assesses the skill of translating a Riemann sum into its corresponding definite integral. The Riemann sum is \(\sum_{i=1}^{40} F(1.5+0.05i)(0.05)\), where \(\Delta x = 0.05\) is the subinterval width. The evaluation points are 1.5 + 0.05i for i from 1 to 40, from 1.55 to 3.5, with 40 terms. This right Riemann sum approximates \(\int_{1.5}^{3.5} F(x),dx\), total length 40 × 0.05 = 2, from 1.5 to 3.5. A tempting distractor like choice B, \(\int_{1.55}^{3.55} F(x),dx\), fails by shifting limits to points without maintaining the interval length. A general strategy for translating Riemann sums is to identify \(\Delta x\), count the terms to find the total length, and determine the limits by aligning the starting point with the overall interval.

This problem tests the skill of translating a Riemann sum into its corresponding definite integral. The factor $\frac{3}{n}$ represents \Delta x, indicating the total interval length is 3. The expression $\left($\frac{3k}{n}$\right)^2$ maps to x_k = k \Delta x, with function $x^2$. The limits are from 0 to 3, as points go from near 0 to 3. A tempting distractor is choice C, \int_${0}^{3}$$(3x)^2$,dx, which fails by incorrectly scaling the integrand with 3 inside. A general strategy for translating Riemann sums is to identify \Delta x = $\frac{b-a}{n}$, the form of x_i = a + i \Delta x, and the function f(x_i), so the sum approximates \int_$a^b$ f(x) dx as n approaches infinity.

This question assesses the skill of translating a Riemann sum into its corresponding definite integral. The Riemann sum is \(\sum_{i=1}^{50} h(5+0.2i)(0.2)\), where \(\Delta x = 0.2\) is the width of each subinterval. The evaluation points are 5 + 0.2i for i from 1 to 50, ranging from 5.2 to 15, with 50 terms. This corresponds to a right Riemann sum for \(\int_{5}^{15} h(x),dx\), with total length 50 × 0.2 = 10, from 5 to 15. A tempting distractor like choice D, \(\int_{5.2}^{15.2} h(x),dx\), fails by shifting both limits to the evaluation points without preserving the original interval length. A general strategy for translating Riemann sums is to identify \(\Delta x\), count the terms to find the total length, and determine the limits by aligning the starting point with the overall interval.

This problem tests the skill of translating a Riemann sum into its corresponding definite integral. The factor $\frac{2}{n}$ represents \Delta x, indicating the total interval length is 2. The exponent $\frac{2i}{n}$ maps to x_i = i \Delta x, with function $e^x$. The limits are from 0 to 2, as the sample points cover near 0 to 2. A tempting distractor is choice D, \int_${0}^{2}$$e^{2x}$,dx, which fails by incorrectly doubling the exponent instead of recognizing the scaling in \Delta x. A general strategy for translating Riemann sums is to identify \Delta x = $\frac{b-a}{n}$, the form of x_i = a + i \Delta x, and the function f(x_i), so the sum approximates \int_$a^b$ f(x) dx as n approaches infinity.

This question assesses the skill of translating a Riemann sum into its corresponding definite integral. The Riemann sum is \(\sum_{i=1}^{20} f(2+0.1i)(0.1)\), where \(\Delta x = 0.1\) represents the width of each subinterval. The evaluation points are 2 + 0.1i for i from 1 to 20, starting at 2.1 and ending at 4, covering 20 subintervals. This setup corresponds to a right Riemann sum approximating the integral \(\int_{2}^{4} f(x),dx\), as the total length is 20 × 0.1 = 2, spanning from 2 to 4. A tempting distractor like choice D, \(\int_{2.1}^{4.1} f(x),dx\), fails because it incorrectly uses the evaluation points as limits without accounting for the full interval covered by the sum. A general strategy for translating Riemann sums is to identify \(\Delta x\), count the terms to find the total length, and determine the limits by aligning the starting point with the overall interval.

This question tests your ability to translate a Riemann sum into its corresponding definite integral notation. The Riemann sum uses $Δx = 4/n$, indicating an interval width of 4. The argument $4i/n$ increases from approximately 0 to 4 as i goes from 1 to n, mapping to the integral limits from 0 to 4. The integrand $x^3$ corresponds directly to the cubed expression in the sum. A tempting distractor is choice B, which fails because using $(4x)^3$ over 0 to 1 scales by $4^3 = 64$ but integrates over width 1, resulting in 64 times the integral $\int_0^1 x^3 , dx$, which is too large. A transferable translation strategy is to identify $Δx$ as $(b - a)/n$, determine a and b from the range of the argument as the index varies from 1 to n, and set the integrand to match the function of that argument.

This question assesses the skill of translating a Riemann sum into its corresponding definite integral. The Riemann sum is \(\sum_{j=1}^{12} p(3+0.5j)(0.5)\), with \(\Delta x = 0.5\) as the subinterval width. The points are 3 + 0.5j for j from 1 to 12, from 3.5 to 9, covering 12 terms. This is a right Riemann sum approximating \(\int_{3}^{9} p(x),dx\), as the total length is 12 × 0.5 = 6, spanning 3 to 9. A tempting distractor like choice C, \(\int_{3.5}^{9.5} p(x),dx\), fails because it adjusts the limits to the points but adds an extra half-width incorrectly. A general strategy for translating Riemann sums is to identify \(\Delta x\), count the terms to find the total length, and determine the limits by aligning the starting point with the overall interval.

This question assesses the skill of translating a Riemann sum into its corresponding definite integral. The Riemann sum is \(\sum_{i=1}^{30} q(-2+0.1i)(0.1)\), with \(\Delta x = 0.1\) as the width. The points are -2 + 0.1i for i from 1 to 30, from -1.9 to 1, with 30 terms. This is a right Riemann sum for \(\int_{-2}^{1} q(x),dx\), total length 30 × 0.1 = 3, from -2 to 1. A tempting distractor like choice B, \(\int_{-1.9}^{1.1} q(x),dx\), fails by shifting limits to points and adding extra width incorrectly. A general strategy for translating Riemann sums is to identify \(\Delta x\), count the terms to find the total length, and determine the limits by aligning the starting point with the overall interval.