Approximating Areas With Riemann Sums - AP Calculus BC
Card 1 of 30
Calculate $L_2$ for $f(x)=x^3$ on $[1,3]$ with $n=2$.
Calculate $L_2$ for $f(x)=x^3$ on $[1,3]$ with $n=2$.
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$L_2 = 9$. $\triangle x = 1$; sum $f(1) + f(2) = 1 + 8 = 9$.
$L_2 = 9$. $\triangle x = 1$; sum $f(1) + f(2) = 1 + 8 = 9$.
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State the formula for the midpoint Riemann sum.
State the formula for the midpoint Riemann sum.
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$M_n = \text{sum of } f(m_i)\times \text{width}$, using midpoints. Each rectangle height uses function value at interval's center.
$M_n = \text{sum of } f(m_i)\times \text{width}$, using midpoints. Each rectangle height uses function value at interval's center.
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What is the effect of increasing $n$ on Riemann sums?
What is the effect of increasing $n$ on Riemann sums?
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Increases accuracy of the approximation. More rectangles provide finer approximation of curved area.
Increases accuracy of the approximation. More rectangles provide finer approximation of curved area.
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Calculate $L_4$ for $f(x)=x$ on $[0,4]$ with $n=4$.
Calculate $L_4$ for $f(x)=x$ on $[0,4]$ with $n=4$.
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$L_4 = 6$. $\triangle x = 1$; sum $f(0) + f(1) + f(2) + f(3) = 0 + 1 + 2 + 3 = 6$.
$L_4 = 6$. $\triangle x = 1$; sum $f(0) + f(1) + f(2) + f(3) = 0 + 1 + 2 + 3 = 6$.
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State the formula for the left Riemann sum.
State the formula for the left Riemann sum.
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$L_n = \text{sum of } f(x_i^*)\times \text{width}$, using left endpoints. Each rectangle height uses function value at left edge of interval.
$L_n = \text{sum of } f(x_i^*)\times \text{width}$, using left endpoints. Each rectangle height uses function value at left edge of interval.
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What is the role of $n$ in a Riemann sum?
What is the role of $n$ in a Riemann sum?
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Number of subintervals for the partition. Controls precision by determining how many subintervals are used.
Number of subintervals for the partition. Controls precision by determining how many subintervals are used.
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What is the trapezoidal rule?
What is the trapezoidal rule?
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An approximation method using trapezoids to estimate area. Connects consecutive points with straight lines to form trapezoids.
An approximation method using trapezoids to estimate area. Connects consecutive points with straight lines to form trapezoids.
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Identify the type of Riemann sum that tends to underestimate.
Identify the type of Riemann sum that tends to underestimate.
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Left Riemann sum (if $f(x)$ is increasing). Left endpoints sample lower function values on increasing functions.
Left Riemann sum (if $f(x)$ is increasing). Left endpoints sample lower function values on increasing functions.
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Calculate $M_4$ for $f(x)=x$ on $[0,4]$ with $n=4$.
Calculate $M_4$ for $f(x)=x$ on $[0,4]$ with $n=4$.
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$M_4 = 8$. $\triangle x = 1$; sum $f(0.5) + f(1.5) + f(2.5) + f(3.5) = 0.5 + 1.5 + 2.5 + 3.5 = 8$.
$M_4 = 8$. $\triangle x = 1$; sum $f(0.5) + f(1.5) + f(2.5) + f(3.5) = 0.5 + 1.5 + 2.5 + 3.5 = 8$.
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Which Riemann sum uses midpoints of intervals?
Which Riemann sum uses midpoints of intervals?
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Midpoint Riemann sum. Uses center point of each subinterval for height calculation.
Midpoint Riemann sum. Uses center point of each subinterval for height calculation.
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Identify the partition width for $n=4$ on $[1,5]$.
Identify the partition width for $n=4$ on $[1,5]$.
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$\triangle x = 1$. $(5-1)/4 = 1$ for the given interval and partitions.
$\triangle x = 1$. $(5-1)/4 = 1$ for the given interval and partitions.
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What is the effect of decreasing $n$ on approximation?
What is the effect of decreasing $n$ on approximation?
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Decreases accuracy of the approximation. Fewer rectangles provide coarser approximation of area.
Decreases accuracy of the approximation. Fewer rectangles provide coarser approximation of area.
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How is a Riemann sum related to the definite integral?
How is a Riemann sum related to the definite integral?
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It approximates the integral as $n \to \text{infinity}$. Riemann sum approaches integral value as partition size approaches zero.
It approximates the integral as $n \to \text{infinity}$. Riemann sum approaches integral value as partition size approaches zero.
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State the formula for the right Riemann sum.
State the formula for the right Riemann sum.
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$R_n = \text{sum of } f(x_i^*)\times \text{width}$, using right endpoints. Each rectangle height uses function value at right edge of interval.
$R_n = \text{sum of } f(x_i^*)\times \text{width}$, using right endpoints. Each rectangle height uses function value at right edge of interval.
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Identify the midpoint for the interval $[3,5]$.
Identify the midpoint for the interval $[3,5]$.
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$m = 4$. Average of interval endpoints: $(3+5)/2 = 4$.
$m = 4$. Average of interval endpoints: $(3+5)/2 = 4$.
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How does a left Riemann sum differ from a right Riemann sum?
How does a left Riemann sum differ from a right Riemann sum?
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Uses left vs. right endpoints of subintervals. Different sampling points within each subinterval affect approximation.
Uses left vs. right endpoints of subintervals. Different sampling points within each subinterval affect approximation.
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What is the general form of a Riemann sum?
What is the general form of a Riemann sum?
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$\text{sum of } f(x_i^*) \times \triangle x$. Standard notation for all Riemann sum variations.
$\text{sum of } f(x_i^*) \times \triangle x$. Standard notation for all Riemann sum variations.
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What is the primary use of Riemann sums?
What is the primary use of Riemann sums?
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To approximate the integral of a function. Estimates area under curves when exact integration is difficult.
To approximate the integral of a function. Estimates area under curves when exact integration is difficult.
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What is necessary for a Riemann sum to equal the exact area?
What is necessary for a Riemann sum to equal the exact area?
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$n \to \text{infinity}$ and $f(x)$ continuous. Infinite partitions with continuous functions guarantee convergence to exact area.
$n \to \text{infinity}$ and $f(x)$ continuous. Infinite partitions with continuous functions guarantee convergence to exact area.
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Calculate $M_2$ for $f(x)=x^2$ on $[0,2]$ with $n=2$.
Calculate $M_2$ for $f(x)=x^2$ on $[0,2]$ with $n=2$.
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$M_2 = 2.5$. $\triangle x = 1$; midpoints $0.5, 1.5$ give $(0.5)^2 + (1.5)^2 = 2.5$.
$M_2 = 2.5$. $\triangle x = 1$; midpoints $0.5, 1.5$ give $(0.5)^2 + (1.5)^2 = 2.5$.
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What does $\triangle x$ represent in Riemann sums?
What does $\triangle x$ represent in Riemann sums?
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Width of each subinterval. Size of each rectangular subdivision in the partition.
Width of each subinterval. Size of each rectangular subdivision in the partition.
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What is the midpoint of the interval $[2,6]$?
What is the midpoint of the interval $[2,6]$?
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$m = 4$. Average of interval endpoints: $(2+6)/2 = 4$.
$m = 4$. Average of interval endpoints: $(2+6)/2 = 4$.
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Identify the type of Riemann sum that tends to overestimate.
Identify the type of Riemann sum that tends to overestimate.
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Right Riemann sum (if $f(x)$ is increasing). Right endpoints sample higher function values on increasing functions.
Right Riemann sum (if $f(x)$ is increasing). Right endpoints sample higher function values on increasing functions.
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How does the trapezoidal rule improve accuracy?
How does the trapezoidal rule improve accuracy?
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Uses trapezoids instead of rectangles. Trapezoids better approximate curved regions than rectangles.
Uses trapezoids instead of rectangles. Trapezoids better approximate curved regions than rectangles.
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Find $\triangle x$ for $n=4$ on $[0,8]$.
Find $\triangle x$ for $n=4$ on $[0,8]$.
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$\triangle x = 2$. $(8-0)/4 = 2$ for the given interval and partitions.
$\triangle x = 2$. $(8-0)/4 = 2$ for the given interval and partitions.
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Calculate $\triangle x$ for $n=5$ on $[2,12]$.
Calculate $\triangle x$ for $n=5$ on $[2,12]$.
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$\triangle x = 2$. $(12-2)/5 = 2$ for the given interval and partitions.
$\triangle x = 2$. $(12-2)/5 = 2$ for the given interval and partitions.
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How is the partition width $\triangle x$ calculated?
How is the partition width $\triangle x$ calculated?
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$\triangle x = \frac{b-a}{n}$, where $[a,b]$ is the interval. Interval length divided by number of subintervals.
$\triangle x = \frac{b-a}{n}$, where $[a,b]$ is the interval. Interval length divided by number of subintervals.
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State the formula for the trapezoidal rule.
State the formula for the trapezoidal rule.
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$T_n = \frac{b-a}{2n} [f(x_0) + 2\text{sum of } f(x_i) + f(x_n)]$. Averages left and right endpoints, summing trapezoid areas.
$T_n = \frac{b-a}{2n} [f(x_0) + 2\text{sum of } f(x_i) + f(x_n)]$. Averages left and right endpoints, summing trapezoid areas.
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What is a Riemann sum?
What is a Riemann sum?
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A method for approximating the total area under a curve. Divides interval into rectangles to estimate area under curve.
A method for approximating the total area under a curve. Divides interval into rectangles to estimate area under curve.
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Calculate $R_4$ for $f(x)=x$ on $[0,4]$ with $n=4$.
Calculate $R_4$ for $f(x)=x$ on $[0,4]$ with $n=4$.
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$R_4 = 10$. $\triangle x = 1$; sum $f(1) + f(2) + f(3) + f(4) = 1 + 2 + 3 + 4 = 10$.
$R_4 = 10$. $\triangle x = 1$; sum $f(1) + f(2) + f(3) + f(4) = 1 + 2 + 3 + 4 = 10$.
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