Calculus Examples

$y = x^{3} + 3$ ; $y = 0$ ; $0 \leq x \leq 2$

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.

$x^{3} + 3 = 0$

Step 1.2

Solve $x^{3} + 3 = 0$ for $x$ .

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Step 1.2.1

Subtract $3$ from both sides of the equation.

$x^{3} = - 3$

Step 1.2.2

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

$x = \sqrt[3]{- 3}$

Step 1.2.3

Simplify $\sqrt[3]{- 3}$ .

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Step 1.2.3.1

Rewrite $- 3$ as ${(- 1)}^{3} \cdot 3$ .

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Step 1.2.3.1.1

Rewrite $- 3$ as $- 1 (3)$ .

$x = \sqrt[3]{- 1 (3)}$

Step 1.2.3.1.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$x = \sqrt[3]{{(- 1)}^{3} \cdot 3}$

Step 1.2.3.2

Pull terms out from under the radical.

$x = - 1 \sqrt[3]{3}$

Step 1.2.3.3

Rewrite $- 1 \sqrt[3]{3}$ as $- \sqrt[3]{3}$ .

$x = - \sqrt[3]{3}$

Step 1.3

Substitute $- \sqrt[3]{3}$ for $x$ .

$y = 0$

Step 1.4

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

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

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.

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

Step 3

Integrate to find the area between $0$ and $2$ .

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Step 3.1

Combine the integrals into a single integral.

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

Step 3.2

Subtract $0$ from $x^{3} + 3$ .

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

Step 3.3

Split the single integral into multiple integrals.

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

Step 3.4

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$\frac{1}{4} x^{4}]_{0}^{2} + \int_{0}^{2} 3 d x$

Step 3.5

Apply the constant rule.

$\frac{1}{4} x^{4}]_{0}^{2} + 3 x]_{0}^{2}$

Step 3.6

Simplify the answer.

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Step 3.6.1

Combine $\frac{1}{4} x^{4}]_{0}^{2}$ and $3 x]_{0}^{2}$ .

$\frac{1}{4} x^{4} + 3 x]_{0}^{2}$

Step 3.6.2

Substitute and simplify.

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Step 3.6.2.1

Evaluate $\frac{1}{4} x^{4} + 3 x$ at $2$ and at $0$ .

$(\frac{1}{4} \cdot 2^{4} + 3 \cdot 2) - (\frac{1}{4} \cdot 0^{4} + 3 \cdot 0)$

Step 3.6.2.2

Simplify.

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Step 3.6.2.2.1

Raise $2$ to the power of $4$ .

$\frac{1}{4} \cdot 16 + 3 \cdot 2 - (\frac{1}{4} \cdot 0^{4} + 3 \cdot 0)$

Step 3.6.2.2.2

Combine $\frac{1}{4}$ and $16$ .

$\frac{16}{4} + 3 \cdot 2 - (\frac{1}{4} \cdot 0^{4} + 3 \cdot 0)$

Step 3.6.2.2.3

Cancel the common factor of $16$ and $4$ .

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Step 3.6.2.2.3.1

Factor $4$ out of $16$ .

$\frac{4 \cdot 4}{4} + 3 \cdot 2 - (\frac{1}{4} \cdot 0^{4} + 3 \cdot 0)$

Step 3.6.2.2.3.2

Cancel the common factors.

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Step 3.6.2.2.3.2.1

Factor $4$ out of $4$ .

$\frac{4 \cdot 4}{4 (1)} + 3 \cdot 2 - (\frac{1}{4} \cdot 0^{4} + 3 \cdot 0)$

Step 3.6.2.2.3.2.2

Cancel the common factor.

$\frac{4 \cdot 4}{4 \cdot 1} + 3 \cdot 2 - (\frac{1}{4} \cdot 0^{4} + 3 \cdot 0)$

Step 3.6.2.2.3.2.3

Rewrite the expression.

$\frac{4}{1} + 3 \cdot 2 - (\frac{1}{4} \cdot 0^{4} + 3 \cdot 0)$

Step 3.6.2.2.3.2.4

Divide $4$ by $1$ .

$4 + 3 \cdot 2 - (\frac{1}{4} \cdot 0^{4} + 3 \cdot 0)$

Step 3.6.2.2.4

Multiply $3$ by $2$ .

$4 + 6 - (\frac{1}{4} \cdot 0^{4} + 3 \cdot 0)$

Step 3.6.2.2.5

Add $4$ and $6$ .

$10 - (\frac{1}{4} \cdot 0^{4} + 3 \cdot 0)$

Step 3.6.2.2.6

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

$10 - (\frac{1}{4} \cdot 0 + 3 \cdot 0)$

Step 3.6.2.2.7

Multiply $\frac{1}{4}$ by $0$ .

$10 - (0 + 3 \cdot 0)$

Step 3.6.2.2.8

Multiply $3$ by $0$ .

$10 - (0 + 0)$

Step 3.6.2.2.9

Add $0$ and $0$ .

$10 - 0$

Step 3.6.2.2.10

Multiply $- 1$ by $0$ .

$10 + 0$

Step 3.6.2.2.11

Add $10$ and $0$ .

$10$

Step 4