Estimate an integral with a left, right or midpoint Riemann sum

\displaystyle\int_a^b f(x)\,dx

1
$\Delta x = \dfrac{b-a}{n}$ $From a table, it is the spacing of the readings you use.$
2
$List the sample points for the sum you were asked for$ $a, a + \Delta x, \dots, b - \Delta x$
3
$f$
4
$\Delta x$ $\displaystyle\sum_{i=1}^{n} f(x_i^*)\,\Delta x$

Remember: Width, sample points, heights, add.

See it in a problem3 worked examples

\displaystyle\int_1^3 \frac{1}{x}\,dx

1
$\Delta x = \dfrac{b-a}{n}$ $\Delta x = \dfrac{3 - 1}{4} = \dfrac{1}{2}$
2
$List the sample points for the sum you were asked for$ $1.5,\ 2,\ 2.5,\ 3$
3
$f$ $\dfrac{2}{3},\ \dfrac{1}{2},\ \dfrac{2}{5},\ \dfrac{1}{3}$
4
$\Delta x$ $\dfrac{1}{2}\left(\dfrac{2}{3} + \dfrac{1}{2} + \dfrac{2}{5} + \dfrac{1}{3}\right) = \dfrac{1}{2} \cdot \dfrac{19}{10} = \dfrac{19}{20}$

\dfrac{19}{20} = 0.95

Common mistakes3

Using $n + 1$ heights. $n$ rectangles use $n$ heights: a left sum skips $b$ , a right sum skips $a$ .
For an increasing $f$ , the left sum is an underestimate and the right sum an overestimate. For a decreasing $f$ , it is the other way around.
Where $f < 0$ , the rectangle's signed area is negative.

2

Evaluate a sum in sigma notation using summation formulas

\displaystyle\sum_{i=1}^{15}\left(i^2 + 4i - 2\right)

1
$Split into constant multiples of standard sums$ $\sum (c\,a_i + b_i) = c\sum a_i + \sum b_i$
2
$Make each sum start where its formula starts$ $\displaystyle\sum_{i=m}^{n} a_i = \sum_{i=1}^{n} a_i - \sum_{i=1}^{m-1} a_i$
3
$Apply the formulas$ $\sum_{i=1}^{n} i = \dfrac{n(n+1)}{2}$
4
$Simplify$

Remember: Split, shift, formula.

See it in a problem3 worked examples

\displaystyle\sum_{i=1}^{12}\left(2i^2 - i + 3\right)

1
$Split into constant multiples of standard sums$ $2\sum_{i=1}^{12} i^2 - \sum_{i=1}^{12} i + \sum_{i=1}^{12} 3$
2
$Make each sum start where its formula starts$ $i = 1$
3
$Apply the formulas$ $2 \cdot \dfrac{12 \cdot 13 \cdot 25}{6} - \dfrac{12 \cdot 13}{2} + 3 \cdot 12 = 2(650) - 78 + 36$
4
$Simplify$ $1300 - 78 + 36 = 1258$

1258

Common mistakes3

Miscounting terms: $\sum_{i=m}^{n}$ has $n - m + 1$ terms, and $\sum_{i=0}^{n} r^i$ has $n + 1$ .
$\sum i^2 \neq \left(\sum i\right)^2$ and $\sum a_i b_i \neq \sum a_i \sum b_i$ .
Writing $\sum_{i=1}^{n} 3 = 3$ . It is $3n$ .

3

Evaluate a definite integral as a limit of Riemann sums

\displaystyle\int_a^b f(x)\,dx

1
$\Delta x = \dfrac{b-a}{n}$
2
$R_n = \displaystyle\sum_{i=1}^{n} f(x_i)\,\Delta x$
3
$i$ $n$
4
$Apply the summation formulas$
5
$n \to \infty$ $n$

Remember: Set up, expand, formulas, limit.

See it in a problem2 worked examples

\displaystyle\int_0^2 \left(3x^2 + 1\right)\,dx

1
$\Delta x = \dfrac{b-a}{n}$ $\Delta x = \dfrac{2}{n}$
2
$R_n = \displaystyle\sum_{i=1}^{n} f(x_i)\,\Delta x$ $R_n = \displaystyle\sum_{i=1}^{n}\left(\frac{12i^2}{n^2} + 1\right)\frac{2}{n}$
3
$i$ $R_n = \dfrac{24}{n^3}\displaystyle\sum_{i=1}^{n} i^2 + \frac{2}{n}\sum_{i=1}^{n} 1$
4
$Apply the summation formulas$ $R_n = \dfrac{24}{n^3} \cdot \dfrac{n(n+1)(2n+1)}{6} + \dfrac{2}{n} \cdot n = \dfrac{4(n+1)(2n+1)}{n^2} + 2$
5
$n \to \infty$ $\displaystyle\lim_{n\to\infty} R_n = 4 \cdot 2 + 2 = 10$

10

Common mistakes3

Pulling $i$ out of the sum. Only factors without $i$ can come out.
$\sum_{i=1}^{n} 1 = n$ , not $1$ .
Forgetting the factor $\Delta x$ . Without it, the sum usually grows without bound.

4

Write a limit of Riemann sums as a definite integral

\displaystyle\lim_{n\to\infty}\sum_{i=1}^{n}\dots

1
$\Delta x$ $\dfrac{i}{n^2} = \dfrac{i}{n}\cdot\dfrac{1}{n}$
2
$x_i = a + i\,\Delta x$
3
$a$
4
$x_i$

Remember: Find the width, find the x, read the ends.

See it in a problem3 worked examples

\displaystyle\lim_{n\to\infty}\sum_{i=1}^{n}\frac{4}{n}\sqrt{1 + \frac{4i}{n}}

1
$\Delta x$ $\Delta x = \dfrac{4}{n}$
2
$x_i = a + i\,\Delta x$ $x_i = 1 + \dfrac{4i}{n} = 1 + i\,\Delta x$
3
$a$ $a = 1$
4
$x_i$ $f(x_i) = \sqrt{x_i}$

\displaystyle\int_1^5 \sqrt{x}\,dx

Common mistakes2

Mismatched pieces: the width $b - a$ must equal $n\,\Delta x$ .
Thinking there is only one answer. Shifting $x$ gives other correct integrals, such as $\int_0^4 \sqrt{1 + x}\,dx = \int_1^5 \sqrt{x}\,dx$ .

5

Evaluate a definite integral using geometry and signed area

e.g. “ Evaluate the following integral by interpreting it as a signed area, and using geometry ”

1
$y = f(x)$ $y = \sqrt{r^2 - x^2}$
2
$Cut the region into triangles, rectangles and circle pieces$
3
$Find each area; below the axis counts negative$
4
$Add the signed areas$

Remember: Sketch, cut, sign, add.

See it in a problem3 worked examples

\displaystyle\int_{-1}^{3}(2 - x)\,dx

1
$y = f(x)$ $(-1, 3)$
2
$Cut the region into triangles, rectangles and circle pieces$ $[-1, 2]$
3
$Find each area; below the axis counts negative$ $\tfrac{1}{2}(3)(3) = \tfrac{9}{2}$
4
$Add the signed areas$ $\tfrac{9}{2} - \tfrac{1}{2} = 4$

4

Common mistakes3

Counting area below the axis as positive. $\int_a^b f(x)\,dx$ is signed area, not total area.
Using a full circle for $\sqrt{r^2 - x^2}$ : it is only the top half.
Treating $|x + 1|$ as a line. It is a V: split it at $x = -1$ .

6

Use properties of definite integrals and odd or even symmetry

\displaystyle\int_a^b f(x)\,dx = \dots

1
$Flip or split the limits until they match$ $\displaystyle\int_b^a f = -\int_a^b f$
2
$Pull out constants and split sums$ $\int (c f + g) = c\int f + \int g$
3
$[-a, a]$ $f(-x) = -f(x)$
4
$Plug in the known values and add$

Remember: Match the limits, split, use symmetry.

See it in a problem3 worked examples

\displaystyle\int_0^5 f(x)\,dx = 4

1
$Flip or split the limits until they match$ $\displaystyle\int_2^5 f = \int_0^5 f - \int_0^2 f = 3$
2
$Pull out constants and split sums$ $\displaystyle\int_2^5 (2f + g) = 2\int_2^5 f + \int_2^5 g$
3
$[-a, a]$ $No symmetric limits here; nothing to test.$
4
$Plug in the known values and add$ $2(3) + (-3) = 3$

3

Common mistakes3

$\displaystyle\int_a^b f(x)g(x)\,dx \neq \int_a^b f(x)\,dx \cdot \int_a^b g(x)\,dx$ .
Calling an integral $0$ because it "looks odd": check $f(-x) = -f(x)$ , and the limits must be $-a$ to $a$ .
Odd times odd is even; odd times even is odd.

The Definite Integral & Riemann Sums

Estimate an integral with a left, right or midpoint Riemann sum

Evaluate a sum in sigma notation using summation formulas

Evaluate a definite integral as a limit of Riemann sums

Write a limit of Riemann sums as a definite integral

Evaluate a definite integral using geometry and signed area

Use properties of definite integrals and odd or even symmetry

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Navigation

Estimate an integral with a left, right or midpoint Riemann sum

Evaluate a sum in sigma notation using summation formulas

Evaluate a definite integral as a limit of Riemann sums

Write a limit of Riemann sums as a definite integral

Evaluate a definite integral using geometry and signed area

Use properties of definite integrals and odd or even symmetry