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Calculus Examples

\sum_{x = 1}^{8} x

$\sum_{x = 1}^{8} x$

Step 1

The formula for the summation of a polynomial with degree

1

$1$ is:

\sum_{k = 1}^{n} k = \frac{n (n + 1)}{2}

$\sum_{k = 1}^{n} k = \frac{n (n + 1)}{2}$

Step 2

Substitute the values into the formula.

\frac{8 (8 + 1)}{2}

$\frac{8 (8 + 1)}{2}$

Step 3

Simplify.

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Step 3.1

Cancel the common factor of

8

$8$ and

2

$2$ .

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Step 3.1.1

Factor

2

$2$ out of

8 (8 + 1)

$8 (8 + 1)$ .

\frac{2 (4 (8 + 1))}{2}

$\frac{2 (4 (8 + 1))}{2}$

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor

2

$2$ out of

2

$2$ .

\frac{2 (4 (8 + 1))}{2 (1)}

$\frac{2 (4 (8 + 1))}{2 (1)}$

Step 3.1.2.2

Cancel the common factor.

\frac{2 (4 (8 + 1))}{2 \cdot 1}

$\frac{2 (4 (8 + 1))}{2 \cdot 1}$

Step 3.1.2.3

Rewrite the expression.

\frac{4 (8 + 1)}{1}

$\frac{4 (8 + 1)}{1}$

Step 3.1.2.4

Divide

4 (8 + 1)

$4 (8 + 1)$ by

1

$1$ .

4 (8 + 1)

$4 (8 + 1)$

4 (8 + 1)

$4 (8 + 1)$

4 (8 + 1)

$4 (8 + 1)$

Step 3.2

Simplify the expression.

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Step 3.2.1

Add

8

$8$ and

1

$1$ .

4 \cdot 9

$4 \cdot 9$

Step 3.2.2

Multiply

4

$4$ by

9

$9$ .

36

$36$

36

$36$

36

$36$

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[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$