Riemann Sums | Math Coach Amy

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Key Ideas Easy Medium Hard

\(\dfrac{\Delta x}{2}(y_0+2y_1+\cdots+y_n)\)

trapezoids with equal widths

What to Remember

Name the sample point

On each subinterval, multiply the width by one height.

A left sum uses the left edge. A right sum uses the right edge. A midpoint sum uses the middle.

Trapezoids average the two edges

For equal widths,

\(\dfrac{\Delta x}{2}\left(y_0 + 2y_1 + \cdots + 2y_{n-1} + y_n\right)\)

That sum is also the average of the left sum and the right sum.

Monotone graphs have a direction

If \(f\) is increasing, the left sum is too small and the right sum is too big.

If \(f\) is decreasing, the left sum is too big and the right sum is too small.

Concavity moves the trapezoids

If the graph is concave up, each chord sits above the curve, so the trapezoidal sum is too big.

If the graph is concave down, the trapezoidal sum is too small. A straight graph makes it exact.

Level 1

Easy

Equal widths. Read the correct column of the table.

Example

Use the table.

x	0	2	4	6
f(x)	4	7	5	3

Approximate \(\int_0^6 f(x)\,dx\) with a right Riemann sum and 3 equal subintervals.

Three equal subintervals on \([0, 6]\).

\(\Delta x = \dfrac{6-0}{3}\)

\(2\)

Right heights: \(f(2)=7\), \(f(4)=5\), \(f(6)=3\).

\(2(7+5+3)\)

\(30\)

Answer: \(30\)

Values of \(g\) are listed.

x	1	3	5	7
g(x)	2	4	6	3

Approximate \(\int_1^7 g(x)\,dx\) with a left Riemann sum and 3 equal subintervals.

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\(\Delta x = \dfrac{7-1}{3}\)

\(2\)

Left heights: \(2\), \(4\), and \(6\).

\(2(2+4+6)\)

\(24\)

Answer: \(24\)

Use a midpoint sum with 2 equal subintervals.

x	0	1	2	3	4
h(x)	2	6	5	3	8

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The interval is \([0, 4]\) and \(\Delta x = 2\).

The midpoints are \(x=1\) and \(x=3\).

\(2\left(h(1)+h(3)\right)\)

\(2(6+3)\)

\(18\)

Answer: \(18\)

\(f\) is positive and increasing on \([0, 6]\). A left Riemann sum with equal subintervals

A. underestimates the integral

B. overestimates the integral

C. equals the integral

D. gives no information about the integral

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On each subinterval the left height is the smallest height.

The rectangles sit under an increasing graph.

The left sum underestimates the integral.

Answer: A. underestimates the integral

Approximate the integral on \([0, 6]\) with the trapezoidal rule and 2 equal subintervals.

x	0	3	6
f(x)	2	8	4

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\(\Delta x = 3\)

\(\dfrac{\Delta x}{2}\left(y_0 + 2y_1 + y_2\right)\)

\(\dfrac{3}{2}\left(2 + 2\cdot 8 + 4\right)\)

\(\dfrac{3}{2}\cdot 22\)

\(33\)

Answer: \(33\)

\(f(x) = 5\) on \([2, 6]\). Find any Riemann sum for \(\int_2^6 f(x)\,dx\) that uses the whole interval as one piece.

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The height is 5 on the whole interval.

\(\Delta x = 6-2\)

\(4\)

\(4\cdot 5\)

\(20\)

Answer: \(20\)

Level 2

Medium

Unequal widths, a straight graph, or a decreasing rate.

Example

The widths are not equal.

x	0	1	4	6
f(x)	2	5	4	1

Approximate \(\int_0^6 f(x)\,dx\) with a right Riemann sum.

Widths: \(1-0=1\), \(4-1=3\), \(6-4=2\).

Right heights: \(5\), \(4\), and \(1\).

\(1\cdot 5 + 3\cdot 4 + 2\cdot 1\)

\(5+12+2\)

\(19\)

Answer: \(19\)

Approximate with a left Riemann sum.

x	0	2	5	6
f(x)	3	1	4	6

Show Answer

Widths: \(2\), \(3\), and \(1\).

Left heights: \(3\), \(1\), and \(4\).

\(2\cdot 3 + 3\cdot 1 + 1\cdot 4\)

\(6+3+4\)

\(13\)

Answer: \(13\)

Let \(f(x)=3x\) on \([1, 5]\). Use the trapezoidal rule with 2 equal subintervals, then explain why it matches \(\int_1^5 3x\,dx\).

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Sample points: \(x=1,3,5\) and \(f=3,9,15\).

\(\dfrac{2}{2}\left(3 + 2\cdot 9 + 15\right)\)

\(36\)

\(\int_1^5 3x\,dx = \left[\dfrac{3}{2}x^2\right]_1^5\)

\(\dfrac{3}{2}(25-1)\)

\(36\)

The graph is a line, so each trapezoid top is the graph. The rule is exact.

Answer: \(36\)

Oil leaks from a tank at a decreasing rate.

t (hours)	0	2	4	6
r(t) (gal/h)	12	9	5	2

Use a right Riemann sum with 3 equal subintervals to approximate the gallons that leak from \(t=0\) to \(t=6\). Is the approximation too big or too small?

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\(\Delta t = 2\)

\(2(9+5+2)\)

\(32\)

The rate is decreasing, so each right height is the smaller height on that piece.

32 gallons is an underestimate.

Answer: 32 gallons, an underestimate

\(f\) is positive and concave up on \([a, b]\). A trapezoidal sum with equal subintervals

A. overestimates the integral

B. underestimates the integral

C. equals the integral

D. must equal the midpoint sum

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Concave up means the chords sit above the graph.

Each trapezoid contains extra area.

The trapezoidal sum overestimates the integral.

Answer: A. overestimates the integral

Which expression is a right Riemann sum for \(\int_2^6 f(x)\,dx\) with 4 equal subintervals?

A. \(f(3)+f(4)+f(5)+f(6)\)

B. \(f(2)+f(3)+f(4)+f(5)\)

C. \(\dfrac{1}{4}\left(f(3)+f(4)+f(5)+f(6)\right)\)

D. \(f(2.5)+f(3.5)+f(4.5)+f(5.5)\)

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\(\Delta x = \dfrac{6-2}{4} = 1\)

The right endpoints are 3, 4, 5, and 6.

The width factor is 1, so the sum is \(f(3)+f(4)+f(5)+f(6)\).

Answer: A. \(f(3)+f(4)+f(5)+f(6)\)

Level 3

Hard

Compare sums, include a negative height, or name the units.

Example

A particle moves on a line with positive decreasing velocity.

t (seconds)	0	2	4	6
v(t) (ft/s)	8	6	3	1

Approximate the distance traveled from \(t=0\) to \(t=6\) with the trapezoidal rule and 3 equal subintervals. Show that this equals the average of the left and right sums.

\(\dfrac{\Delta t}{2}\left(y_0+2y_1+2y_2+y_3\right)\)

\(\dfrac{2}{2}\left(8+2\cdot 6+2\cdot 3+1\right)\)

\(8+12+6+1\)

\(27\)

Left sum: \(2(8+6+3)=34\).

Right sum: \(2(6+3+1)=20\).

\(\dfrac{34+20}{2} = 27\)

Answer: 27 feet

Some heights are negative.

x	1	3	5	7
f(x)	4	\(-2\)	1	\(-3\)

Find the right Riemann sum for \(\int_1^7 f(x)\,dx\) with 3 equal subintervals. What does the sign of the result tell you?

Show Answer

\(\Delta x = 2\)

\(2\left(f(3)+f(5)+f(7)\right)\)

\(2(-2+1-3)\)

\(-8\)

A negative sum approximates signed area. Area below the x-axis counts as negative.

Answer: \(-8\)

Use a midpoint sum with 3 equal subintervals on \([0, 6]\).

x	0	1	2	3	4	5	6
f(x)	2	5	1	4	0	6	3

Show Answer

\(\Delta x = 2\)

Midpoints: \(1\), \(3\), and \(5\).

\(2\left(f(1)+f(3)+f(5)\right)\)

\(2(5+4+6)\)

\(30\)

Answer: \(30\)

\(f\) is strictly increasing and concave down on \([1, 7]\). Put these in order from smallest to largest: the left sum \(L\), the right sum \(R\), the trapezoidal sum \(T\), and the integral \(I\). All three sums use the same equal subintervals.

Show Answer

Increasing: on each piece the left height is smallest and the right height is largest.

\(L < I < R\)

For equal widths, \(T\) is the average of \(L\) and \(R\).

\(L < T < R\)

Concave down: the chords sit under the graph, so \(T < I\).

\(L < T < I < R\)

Answer: \(L < T < I < R\)

\(f\) is increasing on \([0, 4]\). With the same equal partition, the left sum is 11 and the right sum is 17. Which number could equal \(\int_0^4 f(x)\,dx\)?

A. \(9\)

B. \(14\)

C. \(18\)

D. \(20\)

Show Answer

An increasing function traps the integral between the left sum and the right sum.

\(11 < \int_0^4 f(x)\,dx < 17\)

Only 14 sits strictly between 11 and 17.

Answer: B. \(14\)

People enter a museum at the rate below. The rate is not monotone.

t (minutes)	0	5	10	15
r(t) (people/min)	3	7	4	8

Use a right Riemann sum with 3 equal subintervals to approximate how many people enter from \(t=0\) to \(t=15\).

Show Answer

\(\Delta t = 5\)

\(5(7+4+8)\)

\(5\cdot 19\)

\(95\)

The units are people per minute times minutes, so the sum is a number of people.

Answer: 95 people

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