Calculus Examples

$y = \ln (x)$ , $y = x^{2} - 2$

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.

$\ln (x) = x^{2} - 2$

Step 1.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx 0.13793482, 1.56446225 + y = x^{2} - 2$

Step 1.3

Evaluate $y$ when $x = 0.13793482$ .

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Step 1.3.1

Substitute $0.13793482$ for $x$ .

$y = {(0.13793482)}^{2} - 2$

Step 1.3.2

Substitute $0.13793482$ for $x$ in $y = {(0.13793482)}^{2} - 2$ and solve for $y$ .

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Step 1.3.2.1

Remove parentheses.

$y = {0.13793482}^{2} - 2$

Step 1.3.2.2

Remove parentheses.

$y = {(0.13793482)}^{2} - 2$

Step 1.3.2.3

Simplify ${(0.13793482)}^{2} - 2$ .

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Step 1.3.2.3.1

Raise $0.13793482$ to the power of $2$ .

$y = 0.01902601 - 2$

Step 1.3.2.3.2

Subtract $2$ from $0.01902601$ .

$y = - 1.98097398$

Step 1.4

Evaluate $y$ when $x = 1.56446225$ .

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Step 1.4.1

Substitute $1.56446225$ for $x$ .

$y = {(1.56446225)}^{2} - 2$

Step 1.4.2

Substitute $1.56446225$ for $x$ in $y = {(1.56446225)}^{2} - 2$ and solve for $y$ .

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Step 1.4.2.1

Remove parentheses.

$y = {1.56446225}^{2} - 2$

Step 1.4.2.2

Remove parentheses.

$y = {(1.56446225)}^{2} - 2$

Step 1.4.2.3

Simplify ${(1.56446225)}^{2} - 2$ .

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Step 1.4.2.3.1

Raise $1.56446225$ to the power of $2$ .

$y = 2.44754216 - 2$

Step 1.4.2.3.2

Subtract $2$ from $2.44754216$ .

$y = 0.44754216$

Step 1.5

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

$(0.13793482, - 1.98097398)$

$(1.56446225, 0.44754216)$

$(0.13793482, - 1.98097398)$

$(1.56446225, 0.44754216)$

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.

$A r e a = \int_{0.13793482}^{1.56446225} \ln (x) d x - \int_{0.13793482}^{1.56446225} x^{2} - 2 d x$

Step 3

Integrate to find the area between $0.13793482$ and $1.56446225$ .

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Step 3.1

Combine the integrals into a single integral.

$\int_{0.13793482}^{1.56446225} \ln (x) - (x^{2} - 2) d x$

Step 3.2

Simplify each term.

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Step 3.2.1

Apply the distributive property.

$\ln (x) - x^{2} - - 2$

Step 3.2.2

Multiply $- 1$ by $- 2$ .

$\ln (x) - x^{2} + 2$

$\int_{0.13793482}^{1.56446225} \ln (x) - x^{2} + 2 d x$

Step 3.3

Split the single integral into multiple integrals.

$\int_{0.13793482}^{1.56446225} \ln (x) d x + \int_{0.13793482}^{1.56446225} - x^{2} d x + \int_{0.13793482}^{1.56446225} 2 d x$

Step 3.4

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = 1$ .

$\ln (x) x]_{0.13793482}^{1.56446225} - \int_{0.13793482}^{1.56446225} x \frac{1}{x} d x + \int_{0.13793482}^{1.56446225} - x^{2} d x + \int_{0.13793482}^{1.56446225} 2 d x$

Step 3.5

Simplify.

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Step 3.5.1

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

$\ln (x) x]_{0.13793482}^{1.56446225} - \int_{0.13793482}^{1.56446225} \frac{x}{x} d x + \int_{0.13793482}^{1.56446225} - x^{2} d x + \int_{0.13793482}^{1.56446225} 2 d x$

Step 3.5.2

Cancel the common factor of $x$ .

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Step 3.5.2.1

Cancel the common factor.

$\ln (x) x]_{0.13793482}^{1.56446225} - \int_{0.13793482}^{1.56446225} \frac{x}{x} d x + \int_{0.13793482}^{1.56446225} - x^{2} d x + \int_{0.13793482}^{1.56446225} 2 d x$

Step 3.5.2.2

Rewrite the expression.

$\ln (x) x]_{0.13793482}^{1.56446225} - \int_{0.13793482}^{1.56446225} d x + \int_{0.13793482}^{1.56446225} - x^{2} d x + \int_{0.13793482}^{1.56446225} 2 d x$

Step 3.6

Apply the constant rule.

$\ln (x) x]_{0.13793482}^{1.56446225} - (x]_{0.13793482}^{1.56446225}) + \int_{0.13793482}^{1.56446225} - x^{2} d x + \int_{0.13793482}^{1.56446225} 2 d x$

Step 3.7

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

$\ln (x) x]_{0.13793482}^{1.56446225} - (x]_{0.13793482}^{1.56446225}) - \int_{0.13793482}^{1.56446225} x^{2} d x + \int_{0.13793482}^{1.56446225} 2 d x$

Step 3.8

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

$\ln (x) x]_{0.13793482}^{1.56446225} - (x]_{0.13793482}^{1.56446225}) - (\frac{1}{3} x^{3}]_{0.13793482}^{1.56446225}) + \int_{0.13793482}^{1.56446225} 2 d x$

Step 3.9

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

$\ln (x) x]_{0.13793482}^{1.56446225} - (x]_{0.13793482}^{1.56446225}) - (\frac{x^{3}}{3}]_{0.13793482}^{1.56446225}) + \int_{0.13793482}^{1.56446225} 2 d x$

Step 3.10

Apply the constant rule.

$\ln (x) x]_{0.13793482}^{1.56446225} - (x]_{0.13793482}^{1.56446225}) - (\frac{x^{3}}{3}]_{0.13793482}^{1.56446225}) + 2 x]_{0.13793482}^{1.56446225}$

Step 3.11

Simplify the answer.

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Step 3.11.1

Combine $\ln (x) x]_{0.13793482}^{1.56446225}$ and $2 x]_{0.13793482}^{1.56446225}$ .

$\ln (x) x + 2 x]_{0.13793482}^{1.56446225} - (x]_{0.13793482}^{1.56446225}) - (\frac{x^{3}}{3}]_{0.13793482}^{1.56446225})$

Step 3.11.2

Substitute and simplify.

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Step 3.11.2.1

Evaluate $\ln (x) x + 2 x$ at $1.56446225$ and at $0.13793482$ .

$(\ln (1.56446225) \cdot 1.56446225 + 2 \cdot 1.56446225) - (\ln (0.13793482) \cdot 0.13793482 + 2 \cdot 0.13793482) - (x]_{0.13793482}^{1.56446225}) - (\frac{x^{3}}{3}]_{0.13793482}^{1.56446225})$

Step 3.11.2.2

Evaluate $x$ at $1.56446225$ and at $0.13793482$ .

$(\ln (1.56446225) \cdot 1.56446225 + 2 \cdot 1.56446225) - (\ln (0.13793482) \cdot 0.13793482 + 2 \cdot 0.13793482) - ((1.56446225) - 0.13793482) - (\frac{x^{3}}{3}]_{0.13793482}^{1.56446225})$

Step 3.11.2.3

Evaluate $\frac{x^{3}}{3}$ at $1.56446225$ and at $0.13793482$ .

$(\ln (1.56446225) \cdot 1.56446225 + 2 \cdot 1.56446225) - (\ln (0.13793482) \cdot 0.13793482 + 2 \cdot 0.13793482) - ((1.56446225) - 0.13793482) - ((\frac{{1.56446225}^{3}}{3}) - \frac{{0.13793482}^{3}}{3})$

Step 3.11.2.4

Simplify.

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Step 3.11.2.4.1

Multiply $\ln (1.56446225)$ by $1.56446225$ .

$0.70016281 + 2 \cdot 1.56446225 - (\ln (0.13793482) \cdot 0.13793482 + 2 \cdot 0.13793482) - ((1.56446225) - 0.13793482) - ((\frac{{1.56446225}^{3}}{3}) - \frac{{0.13793482}^{3}}{3})$

Step 3.11.2.4.2

Multiply $2$ by $1.56446225$ .

$0.70016281 + 3.12892451 - (\ln (0.13793482) \cdot 0.13793482 + 2 \cdot 0.13793482) - ((1.56446225) - 0.13793482) - ((\frac{{1.56446225}^{3}}{3}) - \frac{{0.13793482}^{3}}{3})$

Step 3.11.2.4.3

Add $0.70016281$ and $3.12892451$ .

$3.82908733 - (\ln (0.13793482) \cdot 0.13793482 + 2 \cdot 0.13793482) - ((1.56446225) - 0.13793482) - ((\frac{{1.56446225}^{3}}{3}) - \frac{{0.13793482}^{3}}{3})$

Step 3.11.2.4.4

Multiply $\ln (0.13793482)$ by $0.13793482$ .

$3.82908733 - (- 0.2732453 + 2 \cdot 0.13793482) - ((1.56446225) - 0.13793482) - ((\frac{{1.56446225}^{3}}{3}) - \frac{{0.13793482}^{3}}{3})$

Step 3.11.2.4.5

Multiply $2$ by $0.13793482$ .

$3.82908733 - (- 0.2732453 + 0.27586965) - ((1.56446225) - 0.13793482) - ((\frac{{1.56446225}^{3}}{3}) - \frac{{0.13793482}^{3}}{3})$

Step 3.11.2.4.6

Add $- 0.2732453$ and $0.27586965$ .

$3.82908733 - 1 \cdot 0.00262435 - ((1.56446225) - 0.13793482) - ((\frac{{1.56446225}^{3}}{3}) - \frac{{0.13793482}^{3}}{3})$

Step 3.11.2.4.7

Multiply $- 1$ by $0.00262435$ .

$3.82908733 - 0.00262435 - ((1.56446225) - 0.13793482) - ((\frac{{1.56446225}^{3}}{3}) - \frac{{0.13793482}^{3}}{3})$

Step 3.11.2.4.8

Subtract $0.00262435$ from $3.82908733$ .

$3.82646298 - ((1.56446225) - 0.13793482) - ((\frac{{1.56446225}^{3}}{3}) - \frac{{0.13793482}^{3}}{3})$

Step 3.11.2.4.9

Subtract $0.13793482$ from $1.56446225$ .

$3.82646298 - 1 \cdot 1.42652743 - ((\frac{{1.56446225}^{3}}{3}) - \frac{{0.13793482}^{3}}{3})$

Step 3.11.2.4.10

Multiply $- 1$ by $1.42652743$ .

$3.82646298 - 1.42652743 - ((\frac{{1.56446225}^{3}}{3}) - \frac{{0.13793482}^{3}}{3})$

Step 3.11.2.4.11

Subtract $1.42652743$ from $3.82646298$ .

$2.39993555 - ((\frac{{1.56446225}^{3}}{3}) - \frac{{0.13793482}^{3}}{3})$

Step 3.11.2.4.12

Raise $1.56446225$ to the power of $3$ .

$2.39993555 - (\frac{3.82908733}{3} - \frac{{0.13793482}^{3}}{3})$

Step 3.11.2.4.13

Raise $0.13793482$ to the power of $3$ .

$2.39993555 - (\frac{3.82908733}{3} - \frac{0.00262435}{3})$

Step 3.11.2.4.14

Combine the numerators over the common denominator.

$2.39993555 - \frac{3.82908733 - 0.00262435}{3}$

Step 3.11.2.4.15

Subtract $0.00262435$ from $3.82908733$ .

$2.39993555 - \frac{3.82646298}{3}$

Step 3.11.2.4.16

To write $2.39993555$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$2.39993555 \cdot \frac{3}{3} - \frac{3.82646298}{3}$

Step 3.11.2.4.17

Combine $2.39993555$ and $\frac{3}{3}$ .

$\frac{2.39993555 \cdot 3}{3} - \frac{3.82646298}{3}$

Step 3.11.2.4.18

Combine the numerators over the common denominator.

$\frac{2.39993555 \cdot 3 - 3.82646298}{3}$

Step 3.11.2.4.19

Multiply $2.39993555$ by $3$ .

$\frac{7.19980666 - 3.82646298}{3}$

Step 3.11.2.4.20

Subtract $3.82646298$ from $7.19980666$ .

$\frac{3.37334367}{3}$

Step 3.12

Divide $3.37334367$ by $3$ .

$1.12444789$

Step 4