Calculus Examples

$\sum_{0}^{10} x^{2} - 2 x + 9$

Step 1

Split the summation to make the starting value of $x$ equal to $1$ .

$\sum_{0}^{10} x^{2} - 2 x + 9 = \sum_{x = 1}^{10} x^{2} - 2 x + 9 + \sum_{x = 0}^{0} x^{2} - 2 x + 9$

Step 2

Evaluate $\sum_{x = 1}^{10} x^{2} - 2 x + 9$ .

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Step 2.1

Split the summation into smaller summations that fit the summation rules.

$\sum_{x = 1}^{10} x^{2} - 2 x + 9 = \sum_{x = 1}^{10} x^{2} - 2 \sum_{x = 1}^{10} x + \sum_{x = 1}^{10} 9$

Step 2.2

Evaluate $\sum_{x = 1}^{10} x^{2}$ .

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Step 2.2.1

The formula for the summation of a polynomial with degree $2$ is:

$\sum_{k = 1}^{n} k^{2} = \frac{n (n + 1) (2 n + 1)}{6}$

Step 2.2.2

Substitute the values into the formula.

$\frac{10 (10 + 1) (2 \cdot 10 + 1)}{6}$

Step 2.2.3

Simplify.

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Step 2.2.3.1

Cancel the common factor of $10$ and $6$ .

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Step 2.2.3.1.1

Factor $2$ out of $10 (10 + 1) (2 \cdot 10 + 1)$ .

$\frac{2 (5 (10 + 1) (2 \cdot 10 + 1))}{6}$

Step 2.2.3.1.2

Cancel the common factors.

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Step 2.2.3.1.2.1

Factor $2$ out of $6$ .

$\frac{2 (5 (10 + 1) (2 \cdot 10 + 1))}{2 (3)}$

Step 2.2.3.1.2.2

Cancel the common factor.

$\frac{2 (5 (10 + 1) (2 \cdot 10 + 1))}{2 \cdot 3}$

Step 2.2.3.1.2.3

Rewrite the expression.

$\frac{5 (10 + 1) (2 \cdot 10 + 1)}{3}$

Step 2.2.3.2

Simplify the numerator.

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Step 2.2.3.2.1

Multiply $2$ by $10$ .

$\frac{5 (10 + 1) (20 + 1)}{3}$

Step 2.2.3.2.2

Add $10$ and $1$ .

$\frac{5 \cdot 11 (20 + 1)}{3}$

Step 2.2.3.2.3

Multiply $5$ by $11$ .

$\frac{55 (20 + 1)}{3}$

Step 2.2.3.2.4

Add $20$ and $1$ .

$\frac{55 \cdot 21}{3}$

Step 2.2.3.3

Simplify the expression.

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Step 2.2.3.3.1

Multiply $55$ by $21$ .

$\frac{1155}{3}$

Step 2.2.3.3.2

Divide $1155$ by $3$ .

$385$

Step 2.3

Evaluate $2 \sum_{x = 1}^{10} x$ .

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Step 2.3.1

The formula for the summation of a polynomial with degree $1$ is:

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

Step 2.3.2

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

$(- 2) (\frac{10 (10 + 1)}{2})$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Simplify the expression.

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Step 2.3.3.1.1

Add $10$ and $1$ .

$- 2 \frac{10 \cdot 11}{2}$

Step 2.3.3.1.2

Multiply $10$ by $11$ .

$- 2 (\frac{110}{2})$

Step 2.3.3.2

Cancel the common factor of $2$ .

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Step 2.3.3.2.1

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{110}{2}$

Step 2.3.3.2.2

Cancel the common factor.

$2 \cdot - 1 \frac{110}{2}$

Step 2.3.3.2.3

Rewrite the expression.

$- 1 \cdot 110$

Step 2.3.3.3

Multiply $- 1$ by $110$ .

$- 110$

Step 2.4

Evaluate $\sum_{x = 1}^{10} 9$ .

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Step 2.4.1

The formula for the summation of a constant is:

$\sum_{k = 1}^{n} c = c n$

Step 2.4.2

Substitute the values into the formula.

$(9) (10)$

Step 2.4.3

Multiply $9$ by $10$ .

$90$

Step 2.5

Add the results of the summations.

$385 - 110 + 90$

Step 2.6

Simplify.

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Step 2.6.1

Subtract $110$ from $385$ .

$275 + 90$

Step 2.6.2

Add $275$ and $90$ .

$365$

Step 3

Evaluate $\sum_{x = 0}^{0} x^{2} - 2 x + 9$ .

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Step 3.1

Expand the series for each value of $x$ .

$0^{2} - 2 \cdot 0 + 9$

Step 3.2

Simplify.

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Step 3.2.1

Raise $0$ to the power of $2$ .

$0 - 2 \cdot 0 + 9$

Step 3.2.2

Multiply $- 2$ by $0$ .

$0 + 0 + 9$

Step 3.2.3

Add $0$ and $0$ .

$0 + 9$

Step 3.2.4

Add $0$ and $9$ .

$9$

Step 4

Replace the summations with the values found.

$365 + 9$

Step 5

Add $365$ and $9$ .

$374$