Calculus Examples

$y = x^{2} - 3$ , $[0, 6]$

Step 1

Solve by substitution to find the intersection between the curves.

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Step 1.1

Eliminate the equal sides of each equation and combine.

$x^{2} - 3 = 0$

Step 1.2

Solve $x^{2} - 3 = 0$ for $x$ .

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Step 1.2.1

Add $3$ to both sides of the equation.

$x^{2} = 3$

Step 1.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{3}$

Step 1.2.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{3}$

Step 1.2.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{3}$

Step 1.2.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{3}, - \sqrt{3}$

Step 1.3

Substitute $\sqrt{3}$ for $x$ .

$y = 0$

Step 1.4

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(\sqrt{3}, 0)$

$(- \sqrt{3}, 0)$

$(\sqrt{3}, 0)$

$(- \sqrt{3}, 0)$

Step 2

The area of the region between the curves is defined as the integral of the upper curve minus the integral of the lower curve over each region. The regions are determined by the intersection points of the curves. This can be done algebraically or graphically.

$A r e a = \int_{0}^{\sqrt{3}} 0 d x - \int_{0}^{\sqrt{3}} x^{2} - 3 d x$

Step 3

Integrate to find the area between $0$ and $\sqrt{3}$ .

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Step 3.1

Combine the integrals into a single integral.

$\int_{0}^{\sqrt{3}} 0 - (x^{2} - 3) d x$

Step 3.2

Subtract $x^{2} - 3$ from $0$ .

$\int_{0}^{\sqrt{3}} - (x^{2} - 3) d x$

Step 3.3

Apply the distributive property.

$\int_{0}^{\sqrt{3}} - x^{2} + 3 d x$

Step 3.4

Split the single integral into multiple integrals.

$\int_{0}^{\sqrt{3}} - x^{2} d x + \int_{0}^{\sqrt{3}} 3 d x$

Step 3.5

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$- \int_{0}^{\sqrt{3}} x^{2} d x + \int_{0}^{\sqrt{3}} 3 d x$

Step 3.6

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$- (\frac{1}{3} x^{3}]_{0}^{\sqrt{3}}) + \int_{0}^{\sqrt{3}} 3 d x$

Step 3.7

Combine $\frac{1}{3}$ and $x^{3}$ .

$- (\frac{x^{3}}{3}]_{0}^{\sqrt{3}}) + \int_{0}^{\sqrt{3}} 3 d x$

Step 3.8

Apply the constant rule.

$- (\frac{x^{3}}{3}]_{0}^{\sqrt{3}}) + 3 x]_{0}^{\sqrt{3}}$

Step 3.9

Simplify the answer.

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Step 3.9.1

Substitute and simplify.

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Step 3.9.1.1

Evaluate $\frac{x^{3}}{3}$ at $\sqrt{3}$ and at $0$ .

$- ((\frac{{\sqrt{3}}^{3}}{3}) - \frac{0^{3}}{3}) + 3 x]_{0}^{\sqrt{3}}$

Step 3.9.1.2

Evaluate $3 x$ at $\sqrt{3}$ and at $0$ .

$- (\frac{{\sqrt{3}}^{3}}{3} - \frac{0^{3}}{3}) + 3 \sqrt{3} - 3 \cdot 0$

Step 3.9.1.3

Simplify.

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Step 3.9.1.3.1

Rewrite ${\sqrt{3}}^{3}$ as $\sqrt{3^{3}}$ .

$- (\frac{\sqrt{3^{3}}}{3} - \frac{0^{3}}{3}) + 3 \sqrt{3} - 3 \cdot 0$

Step 3.9.1.3.2

Raise $3$ to the power of $3$ .

$- (\frac{\sqrt{27}}{3} - \frac{0^{3}}{3}) + 3 \sqrt{3} - 3 \cdot 0$

Step 3.9.1.3.3

Raising $0$ to any positive power yields $0$ .

$- (\frac{\sqrt{27}}{3} - \frac{0}{3}) + 3 \sqrt{3} - 3 \cdot 0$

Step 3.9.1.3.4

Cancel the common factor of $0$ and $3$ .

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Step 3.9.1.3.4.1

Factor $3$ out of $0$ .

$- (\frac{\sqrt{27}}{3} - \frac{3 (0)}{3}) + 3 \sqrt{3} - 3 \cdot 0$

Step 3.9.1.3.4.2

Cancel the common factors.

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Step 3.9.1.3.4.2.1

Factor $3$ out of $3$ .

$- (\frac{\sqrt{27}}{3} - \frac{3 \cdot 0}{3 \cdot 1}) + 3 \sqrt{3} - 3 \cdot 0$

Step 3.9.1.3.4.2.2

Cancel the common factor.

$- (\frac{\sqrt{27}}{3} - \frac{3 \cdot 0}{3 \cdot 1}) + 3 \sqrt{3} - 3 \cdot 0$

Step 3.9.1.3.4.2.3

Rewrite the expression.

$- (\frac{\sqrt{27}}{3} - \frac{0}{1}) + 3 \sqrt{3} - 3 \cdot 0$

Step 3.9.1.3.4.2.4

Divide $0$ by $1$ .

$- (\frac{\sqrt{27}}{3} - 0) + 3 \sqrt{3} - 3 \cdot 0$

Step 3.9.1.3.5

Multiply $- 1$ by $0$ .

$- (\frac{\sqrt{27}}{3} + 0) + 3 \sqrt{3} - 3 \cdot 0$

Step 3.9.1.3.6

Add $\frac{\sqrt{27}}{3}$ and $0$ .

$- \frac{\sqrt{27}}{3} + 3 \sqrt{3} - 3 \cdot 0$

Step 3.9.1.3.7

Multiply $- 3$ by $0$ .

$- \frac{\sqrt{27}}{3} + 3 \sqrt{3} + 0$

Step 3.9.1.3.8

Add $3 \sqrt{3}$ and $0$ .

$- \frac{\sqrt{27}}{3} + 3 \sqrt{3}$

Step 3.9.2

Simplify.

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Step 3.9.2.1

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 3.9.2.1.1

Factor $9$ out of $27$ .

$- \frac{\sqrt{9 (3)}}{3} + 3 \sqrt{3}$

Step 3.9.2.1.2

Rewrite $9$ as $3^{2}$ .

$- \frac{\sqrt{3^{2} \cdot 3}}{3} + 3 \sqrt{3}$

Step 3.9.2.2

Pull terms out from under the radical.

$- \frac{3 \sqrt{3}}{3} + 3 \sqrt{3}$

Step 3.9.2.3

Cancel the common factor of $3$ .

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Step 3.9.2.3.1

Cancel the common factor.

$- \frac{3 \sqrt{3}}{3} + 3 \sqrt{3}$

Step 3.9.2.3.2

Divide $\sqrt{3}$ by $1$ .

$- \sqrt{3} + 3 \sqrt{3}$

Step 3.9.2.4

Add $- \sqrt{3}$ and $3 \sqrt{3}$ .

$2 \sqrt{3}$

Step 4

$A r e a = \int_{\sqrt{3}}^{6} x^{2} - 3 d x - \int_{\sqrt{3}}^{6} 0 d x$

Step 5

Integrate to find the area between $\sqrt{3}$ and $6$ .

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Step 5.1

Combine the integrals into a single integral.

$\int_{\sqrt{3}}^{6} x^{2} - 3 - (0) d x$

Step 5.2

Subtract $0$ from $x^{2} - 3$ .

$\int_{\sqrt{3}}^{6} x^{2} - 3 d x$

Step 5.3

Split the single integral into multiple integrals.

$\int_{\sqrt{3}}^{6} x^{2} d x + \int_{\sqrt{3}}^{6} - 3 d x$

Step 5.4

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$\frac{1}{3} x^{3}]_{\sqrt{3}}^{6} + \int_{\sqrt{3}}^{6} - 3 d x$

Step 5.5

Apply the constant rule.

$\frac{1}{3} x^{3}]_{\sqrt{3}}^{6} + - 3 x]_{\sqrt{3}}^{6}$

Step 5.6

Simplify the answer.

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Step 5.6.1

Combine $\frac{1}{3} x^{3}]_{\sqrt{3}}^{6}$ and $- 3 x]_{\sqrt{3}}^{6}$ .

$\frac{1}{3} x^{3} - 3 x]_{\sqrt{3}}^{6}$

Step 5.6.2

Substitute and simplify.

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Step 5.6.2.1

Evaluate $\frac{1}{3} x^{3} - 3 x$ at $6$ and at $\sqrt{3}$ .

$(\frac{1}{3} \cdot 6^{3} - 3 \cdot 6) - (\frac{1}{3} {\sqrt{3}}^{3} - 3 \sqrt{3})$

Step 5.6.2.2

Simplify.

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Step 5.6.2.2.1

Raise $6$ to the power of $3$ .

$\frac{1}{3} \cdot 216 - 3 \cdot 6 - (\frac{1}{3} {\sqrt{3}}^{3} - 3 \sqrt{3})$

Step 5.6.2.2.2

Combine $\frac{1}{3}$ and $216$ .

$\frac{216}{3} - 3 \cdot 6 - (\frac{1}{3} {\sqrt{3}}^{3} - 3 \sqrt{3})$

Step 5.6.2.2.3

Cancel the common factor of $216$ and $3$ .

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Step 5.6.2.2.3.1

Factor $3$ out of $216$ .

$\frac{3 \cdot 72}{3} - 3 \cdot 6 - (\frac{1}{3} {\sqrt{3}}^{3} - 3 \sqrt{3})$

Step 5.6.2.2.3.2

Cancel the common factors.

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Step 5.6.2.2.3.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 72}{3 (1)} - 3 \cdot 6 - (\frac{1}{3} {\sqrt{3}}^{3} - 3 \sqrt{3})$

Step 5.6.2.2.3.2.2

Cancel the common factor.

$\frac{3 \cdot 72}{3 \cdot 1} - 3 \cdot 6 - (\frac{1}{3} {\sqrt{3}}^{3} - 3 \sqrt{3})$

Step 5.6.2.2.3.2.3

Rewrite the expression.

$\frac{72}{1} - 3 \cdot 6 - (\frac{1}{3} {\sqrt{3}}^{3} - 3 \sqrt{3})$

Step 5.6.2.2.3.2.4

Divide $72$ by $1$ .

$72 - 3 \cdot 6 - (\frac{1}{3} {\sqrt{3}}^{3} - 3 \sqrt{3})$

Step 5.6.2.2.4

Multiply $- 3$ by $6$ .

$72 - 18 - (\frac{1}{3} {\sqrt{3}}^{3} - 3 \sqrt{3})$

Step 5.6.2.2.5

Subtract $18$ from $72$ .

$54 - (\frac{1}{3} {\sqrt{3}}^{3} - 3 \sqrt{3})$

Step 5.6.2.2.6

Rewrite ${\sqrt{3}}^{3}$ as $\sqrt{3^{3}}$ .

$54 - (\frac{1}{3} \sqrt{3^{3}} - 3 \sqrt{3})$

Step 5.6.2.2.7

Raise $3$ to the power of $3$ .

$54 - (\frac{1}{3} \sqrt{27} - 3 \sqrt{3})$

Step 5.6.2.2.8

Combine $\frac{1}{3}$ and $\sqrt{27}$ .

$54 - (\frac{\sqrt{27}}{3} - 3 \sqrt{3})$

Step 5.6.3

Simplify.

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Step 5.6.3.1

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 5.6.3.1.1

Factor $9$ out of $27$ .

$54 - (\frac{\sqrt{9 (3)}}{3} - 3 \sqrt{3})$

Step 5.6.3.1.2

Rewrite $9$ as $3^{2}$ .

$54 - (\frac{\sqrt{3^{2} \cdot 3}}{3} - 3 \sqrt{3})$

Step 5.6.3.2

Pull terms out from under the radical.

$54 - (\frac{3 \sqrt{3}}{3} - 3 \sqrt{3})$

Step 5.6.3.3

Cancel the common factor of $3$ .

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Step 5.6.3.3.1

Cancel the common factor.

$54 - (\frac{3 \sqrt{3}}{3} - 3 \sqrt{3})$

Step 5.6.3.3.2

Divide $\sqrt{3}$ by $1$ .

$54 - (\sqrt{3} - 3 \sqrt{3})$

Step 5.6.3.4

Subtract $3 \sqrt{3}$ from $\sqrt{3}$ .

$54 - (- 2 \sqrt{3})$

Step 5.6.3.5

Multiply $- 2$ by $- 1$ .

$54 + 2 \sqrt{3}$

Step 6

Add $2 \sqrt{3}$ and $2 \sqrt{3}$ .

$54 + 4 \sqrt{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$54 + 4 \sqrt{3}$

Decimal Form:

$60.92820323 \dots$

Step 8