Calculus Examples

$y = \sec^{2} (x)$ , $0 \leq x \leq \frac{π}{6}$

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.

$\sec^{2} (x) = 0$

Step 1.2

Solve $\sec^{2} (x) = 0$ for $x$ .

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Step 1.2.1

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

$\sec (x) = \pm \sqrt{0}$

Step 1.2.2

Simplify $\pm \sqrt{0}$ .

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Step 1.2.2.1

Rewrite $0$ as $0^{2}$ .

$\sec (x) = \pm \sqrt{0^{2}}$

Step 1.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$\sec (x) = \pm 0$

Step 1.2.2.3

Plus or minus $0$ is $0$ .

$\sec (x) = 0$

Step 1.2.3

The range of secant is $y \leq - 1$ and $y \geq 1$ . Since $0$ does not fall in this range, there is no solution.

No solution

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.

$A r e a = \int_{0}^{\frac{π}{6}} \sec^{2} (x) d x - \int_{0}^{\frac{π}{6}} 0 d x$

Step 3

Integrate to find the area between $0$ and $\frac{π}{6}$ .

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Step 3.1

Combine the integrals into a single integral.

$\int_{0}^{\frac{π}{6}} \sec^{2} (x) - (0) d x$

Step 3.2

Subtract $0$ from $\sec^{2} (x)$ .

$\int_{0}^{\frac{π}{6}} \sec^{2} (x) d x$

Step 3.3

Since the derivative of $\tan (x)$ is $\sec^{2} (x)$ , the integral of $\sec^{2} (x)$ is $\tan (x)$ .

$\tan (x)]_{0}^{\frac{π}{6}}$

Step 3.4

Simplify the answer.

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Step 3.4.1

Evaluate $\tan (x)$ at $\frac{π}{6}$ and at $0$ .

$\tan (\frac{π}{6}) - \tan (0)$

Step 3.4.2

Simplify.

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Step 3.4.2.1

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

$\frac{\sqrt{3}}{3} - \tan (0)$

Step 3.4.2.2

The exact value of $\tan (0)$ is $0$ .

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

Step 3.4.2.3

Multiply $- 1$ by $0$ .

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

Step 3.4.2.4

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

$\frac{\sqrt{3}}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{3}}{3}$

Decimal Form:

$0.57735026 \dots$

Step 5