Calculus Examples

$\sum_{i = 1}^{22} \frac{i^{3}}{3}$

Step 1

Factor $\frac{1}{3}$ out of the summation.

$\frac{1}{3} \sum_{k = 1}^{22} i^{3}$

Step 2

The formula for the summation of a polynomial with degree $3$ is:

$\sum_{k = 1}^{n} i^{3} = \frac{n^{2} {(n + 1)}^{2}}{4}$

Step 3

Substitute the values into the formula and make sure to multiply by the front term.

$(\frac{1}{3}) (\frac{22^{2} {(22 + 1)}^{2}}{4})$

Step 4

Simplify.

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Step 4.1

Combine fractions.

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Step 4.1.1

Combine.

$\frac{1 (22^{2} {(22 + 1)}^{2})}{3 \cdot 4}$

Step 4.1.2

Multiply.

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Step 4.1.2.1

Multiply $22^{2}$ by $1$ .

$\frac{22^{2} {(22 + 1)}^{2}}{3 \cdot 4}$

Step 4.1.2.2

Multiply $3$ by $4$ .

$\frac{22^{2} {(22 + 1)}^{2}}{12}$

Step 4.2

Simplify the numerator.

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Step 4.2.1

Add $22$ and $1$ .

$\frac{22^{2} \cdot 23^{2}}{12}$

Step 4.2.2

Raise $22$ to the power of $2$ .

$\frac{484 \cdot 23^{2}}{12}$

Step 4.2.3

Raise $23$ to the power of $2$ .

$\frac{484 \cdot 529}{12}$

Step 4.3

Reduce the expression by cancelling the common factors.

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Step 4.3.1

Multiply $484$ by $529$ .

$\frac{256036}{12}$

Step 4.3.2

Cancel the common factor of $256036$ and $12$ .

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Step 4.3.2.1

Factor $4$ out of $256036$ .

$\frac{4 (64009)}{12}$

Step 4.3.2.2

Cancel the common factors.

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Step 4.3.2.2.1

Factor $4$ out of $12$ .

$\frac{4 \cdot 64009}{4 \cdot 3}$

Step 4.3.2.2.2

Cancel the common factor.

$\frac{4 \cdot 64009}{4 \cdot 3}$

Step 4.3.2.2.3

Rewrite the expression.

$\frac{64009}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{64009}{3}$

Decimal Form:

$21336.\overline{3}$

Mixed Number Form:

$21336 \frac{1}{3}$