Calculus Examples

$\int_{5}^{10} \sqrt{100 - x^{2}} d x$

Step 1

Let $x = 10 \sin (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = 10 \cos (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $10 \cos (t)$ is positive.

$\int_{\frac{π}{6}}^{\frac{π}{2}} \sqrt{100 - {(10 \sin (t))}^{2}} (10 \cos (t)) d t$

Step 2

Simplify terms.

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Step 2.1

Simplify $\sqrt{100 - {(10 \sin (t))}^{2}}$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the product rule to $10 \sin (t)$ .

$\int_{\frac{π}{6}}^{\frac{π}{2}} \sqrt{100 - (10^{2} \sin^{2} (t))} (10 \cos (t)) d t$

Step 2.1.1.2

Raise $10$ to the power of $2$ .

$\int_{\frac{π}{6}}^{\frac{π}{2}} \sqrt{100 - (100 \sin^{2} (t))} (10 \cos (t)) d t$

Step 2.1.1.3

Multiply $100$ by $- 1$ .

$\int_{\frac{π}{6}}^{\frac{π}{2}} \sqrt{100 - 100 \sin^{2} (t)} (10 \cos (t)) d t$

Step 2.1.2

Factor $100$ out of $100$ .

$\int_{\frac{π}{6}}^{\frac{π}{2}} \sqrt{100 (1) - 100 \sin^{2} (t)} (10 \cos (t)) d t$

Step 2.1.3

Factor $100$ out of $- 100 \sin^{2} (t)$ .

$\int_{\frac{π}{6}}^{\frac{π}{2}} \sqrt{100 (1) + 100 (- \sin^{2} (t))} (10 \cos (t)) d t$

Step 2.1.4

Factor $100$ out of $100 (1) + 100 (- \sin^{2} (t))$ .

$\int_{\frac{π}{6}}^{\frac{π}{2}} \sqrt{100 (1 - \sin^{2} (t))} (10 \cos (t)) d t$

Step 2.1.5

Apply pythagorean identity.

$\int_{\frac{π}{6}}^{\frac{π}{2}} \sqrt{100 \cos^{2} (t)} (10 \cos (t)) d t$

Step 2.1.6

Rewrite $100 \cos^{2} (t)$ as ${(10 \cos (t))}^{2}$ .

$\int_{\frac{π}{6}}^{\frac{π}{2}} \sqrt{{(10 \cos (t))}^{2}} (10 \cos (t)) d t$

Step 2.1.7

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

$\int_{\frac{π}{6}}^{\frac{π}{2}} 10 \cos (t) (10 \cos (t)) d t$

Step 2.2

Simplify.

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Step 2.2.1

Multiply $10$ by $10$ .

$\int_{\frac{π}{6}}^{\frac{π}{2}} 100 \cos (t) \cos (t) d t$

Step 2.2.2

Raise $\cos (t)$ to the power of $1$ .

$\int_{\frac{π}{6}}^{\frac{π}{2}} 100 (\cos^{1} (t) \cos (t)) d t$

Step 2.2.3

Raise $\cos (t)$ to the power of $1$ .

$\int_{\frac{π}{6}}^{\frac{π}{2}} 100 (\cos^{1} (t) \cos^{1} (t)) d t$

Step 2.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int_{\frac{π}{6}}^{\frac{π}{2}} 100 {\cos (t)}^{1 + 1} d t$

Step 2.2.5

Add $1$ and $1$ .

$\int_{\frac{π}{6}}^{\frac{π}{2}} 100 \cos^{2} (t) d t$

Step 3

Since $100$ is constant with respect to $t$ , move $100$ out of the integral.

$100 \int_{\frac{π}{6}}^{\frac{π}{2}} \cos^{2} (t) d t$

Step 4

Use the half-angle formula to rewrite $\cos^{2} (t)$ as $\frac{1 + \cos (2 t)}{2}$ .

$100 \int_{\frac{π}{6}}^{\frac{π}{2}} \frac{1 + \cos (2 t)}{2} d t$

Step 5

Since $\frac{1}{2}$ is constant with respect to $t$ , move $\frac{1}{2}$ out of the integral.

$100 (\frac{1}{2} \int_{\frac{π}{6}}^{\frac{π}{2}} 1 + \cos (2 t) d t)$

Step 6

Simplify.

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Step 6.1

Combine $\frac{1}{2}$ and $100$ .

$\frac{100}{2} \int_{\frac{π}{6}}^{\frac{π}{2}} 1 + \cos (2 t) d t$

Step 6.2

Cancel the common factor of $100$ and $2$ .

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Step 6.2.1

Factor $2$ out of $100$ .

$\frac{2 \cdot 50}{2} \int_{\frac{π}{6}}^{\frac{π}{2}} 1 + \cos (2 t) d t$

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 50}{2 (1)} \int_{\frac{π}{6}}^{\frac{π}{2}} 1 + \cos (2 t) d t$

Step 6.2.2.2

Cancel the common factor.

$\frac{2 \cdot 50}{2 \cdot 1} \int_{\frac{π}{6}}^{\frac{π}{2}} 1 + \cos (2 t) d t$

Step 6.2.2.3

Rewrite the expression.

$\frac{50}{1} \int_{\frac{π}{6}}^{\frac{π}{2}} 1 + \cos (2 t) d t$

Step 6.2.2.4

Divide $50$ by $1$ .

$50 \int_{\frac{π}{6}}^{\frac{π}{2}} 1 + \cos (2 t) d t$

Step 7

Split the single integral into multiple integrals.

$50 (\int_{\frac{π}{6}}^{\frac{π}{2}} d t + \int_{\frac{π}{6}}^{\frac{π}{2}} \cos (2 t) d t)$

Step 8

Apply the constant rule.

$50 (t]_{\frac{π}{6}}^{\frac{π}{2}} + \int_{\frac{π}{6}}^{\frac{π}{2}} \cos (2 t) d t)$

Step 9

Let $u = 2 t$ . Then $d u = 2 d t$ , so $\frac{1}{2} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 9.1

Let $u = 2 t$ . Find $\frac{d u}{d t}$ .

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Step 9.1.1

Differentiate $2 t$ .

$\frac{d}{d t} [2 t]$

Step 9.1.2

Since $2$ is constant with respect to $t$ , the derivative of $2 t$ with respect to $t$ is $2 \frac{d}{d t} [t]$ .

$2 \frac{d}{d t} [t]$

Step 9.1.3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 9.1.4

Multiply $2$ by $1$ .

$2$

Step 9.2

Substitute the lower limit in for $t$ in $u = 2 t$ .

$u_{lower} = 2 \frac{π}{6}$

Step 9.3

Cancel the common factor of $2$ .

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Step 9.3.1

Factor $2$ out of $6$ .

$u_{lower} = 2 \frac{π}{2 (3)}$

Step 9.3.2

Cancel the common factor.

$u_{lower} = 2 \frac{π}{2 \cdot 3}$

Step 9.3.3

Rewrite the expression.

$u_{lower} = \frac{π}{3}$

Step 9.4

Substitute the upper limit in for $t$ in $u = 2 t$ .

$u_{upper} = 2 \frac{π}{2}$

Step 9.5

Cancel the common factor of $2$ .

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Step 9.5.1

Cancel the common factor.

$u_{upper} = 2 \frac{π}{2}$

Step 9.5.2

Rewrite the expression.

$u_{upper} = π$

Step 9.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = \frac{π}{3}$

$u_{upper} = π$

Step 9.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$50 (t]_{\frac{π}{6}}^{\frac{π}{2}} + \int_{\frac{π}{3}}^{π} \cos (u) \frac{1}{2} d u)$

Step 10

Combine $\cos (u)$ and $\frac{1}{2}$ .

$50 (t]_{\frac{π}{6}}^{\frac{π}{2}} + \int_{\frac{π}{3}}^{π} \frac{\cos (u)}{2} d u)$

Step 11

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$50 (t]_{\frac{π}{6}}^{\frac{π}{2}} + \frac{1}{2} \int_{\frac{π}{3}}^{π} \cos (u) d u)$

Step 12

The integral of $\cos (u)$ with respect to $u$ is $\sin (u)$ .

$50 (t]_{\frac{π}{6}}^{\frac{π}{2}} + \frac{1}{2} \sin (u)]_{\frac{π}{3}}^{π})$

Step 13

Combine $\frac{1}{2}$ and $\sin (u)]_{\frac{π}{3}}^{π}$ .

$50 (t]_{\frac{π}{6}}^{\frac{π}{2}} + \frac{\sin (u)]_{\frac{π}{3}}^{π}}{2})$

Step 14

Substitute and simplify.

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Step 14.1

Evaluate $t$ at $\frac{π}{2}$ and at $\frac{π}{6}$ .

$50 ((\frac{π}{2}) - \frac{π}{6} + \frac{\sin (u)]_{\frac{π}{3}}^{π}}{2})$

Step 14.2

Evaluate $\sin (u)$ at $π$ and at $\frac{π}{3}$ .

$50 ((\frac{π}{2}) - \frac{π}{6} + \frac{(\sin (π)) - \sin (\frac{π}{3})}{2})$

Step 14.3

Simplify.

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Step 14.3.1

To write $\frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$50 (\frac{π}{2} \cdot \frac{3}{3} - \frac{π}{6} + \frac{(\sin (π)) - \sin (\frac{π}{3})}{2})$

Step 14.3.2

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 14.3.2.1

Multiply $\frac{π}{2}$ by $\frac{3}{3}$ .

$50 (\frac{π \cdot 3}{2 \cdot 3} - \frac{π}{6} + \frac{(\sin (π)) - \sin (\frac{π}{3})}{2})$

Step 14.3.2.2

Multiply $2$ by $3$ .

$50 (\frac{π \cdot 3}{6} - \frac{π}{6} + \frac{(\sin (π)) - \sin (\frac{π}{3})}{2})$

Step 14.3.3

Combine the numerators over the common denominator.

$50 (\frac{π \cdot 3 - π}{6} + \frac{(\sin (π)) - \sin (\frac{π}{3})}{2})$

Step 14.3.4

Move $3$ to the left of $π$ .

$50 (\frac{3 \cdot π - π}{6} + \frac{(\sin (π)) - \sin (\frac{π}{3})}{2})$

Step 14.3.5

Subtract $π$ from $3 π$ .

$50 (\frac{2 π}{6} + \frac{(\sin (π)) - \sin (\frac{π}{3})}{2})$

Step 14.3.6

Cancel the common factor of $2$ and $6$ .

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Step 14.3.6.1

Factor $2$ out of $2 π$ .

$50 (\frac{2 (π)}{6} + \frac{(\sin (π)) - \sin (\frac{π}{3})}{2})$

Step 14.3.6.2

Cancel the common factors.

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Step 14.3.6.2.1

Factor $2$ out of $6$ .

$50 (\frac{2 π}{2 \cdot 3} + \frac{(\sin (π)) - \sin (\frac{π}{3})}{2})$

Step 14.3.6.2.2

Cancel the common factor.

$50 (\frac{2 π}{2 \cdot 3} + \frac{(\sin (π)) - \sin (\frac{π}{3})}{2})$

Step 14.3.6.2.3

Rewrite the expression.

$50 (\frac{π}{3} + \frac{(\sin (π)) - \sin (\frac{π}{3})}{2})$

$50 (\frac{π}{3} + \frac{\sin (π) - \sin (\frac{π}{3})}{2})$

Step 15

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$50 (\frac{π}{3} + \frac{\sin (π) - \frac{\sqrt{3}}{2}}{2})$

Step 16

Simplify.

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Step 16.1

Simplify each term.

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Step 16.1.1

Simplify the numerator.

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Step 16.1.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$50 (\frac{π}{3} + \frac{\sin (0) - \frac{\sqrt{3}}{2}}{2})$

Step 16.1.1.2

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

$50 (\frac{π}{3} + \frac{0 - \frac{\sqrt{3}}{2}}{2})$

Step 16.1.1.3

Subtract $\frac{\sqrt{3}}{2}$ from $0$ .

$50 (\frac{π}{3} + \frac{- \frac{\sqrt{3}}{2}}{2})$

Step 16.1.2

Multiply the numerator by the reciprocal of the denominator.

$50 (\frac{π}{3} - \frac{\sqrt{3}}{2} \cdot \frac{1}{2})$

Step 16.1.3

Multiply $- \frac{\sqrt{3}}{2} \cdot \frac{1}{2}$ .

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Step 16.1.3.1

Multiply $\frac{1}{2}$ by $\frac{\sqrt{3}}{2}$ .

$50 (\frac{π}{3} - \frac{\sqrt{3}}{2 \cdot 2})$

Step 16.1.3.2

Multiply $2$ by $2$ .

$50 (\frac{π}{3} - \frac{\sqrt{3}}{4})$

Step 16.2

Apply the distributive property.

$50 \frac{π}{3} + 50 (- \frac{\sqrt{3}}{4})$

Step 16.3

Combine $50$ and $\frac{π}{3}$ .

$\frac{50 π}{3} + 50 (- \frac{\sqrt{3}}{4})$

Step 16.4

Cancel the common factor of $2$ .

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Step 16.4.1

Move the leading negative in $- \frac{\sqrt{3}}{4}$ into the numerator.

$\frac{50 π}{3} + 50 \frac{- \sqrt{3}}{4}$

Step 16.4.2

Factor $2$ out of $50$ .

$\frac{50 π}{3} + 2 (25) \frac{- \sqrt{3}}{4}$

Step 16.4.3

Factor $2$ out of $4$ .

$\frac{50 π}{3} + 2 \cdot 25 \frac{- \sqrt{3}}{2 \cdot 2}$

Step 16.4.4

Cancel the common factor.

$\frac{50 π}{3} + 2 \cdot 25 \frac{- \sqrt{3}}{2 \cdot 2}$

Step 16.4.5

Rewrite the expression.

$\frac{50 π}{3} + 25 \frac{- \sqrt{3}}{2}$

Step 16.5

Combine $25$ and $\frac{- \sqrt{3}}{2}$ .

$\frac{50 π}{3} + \frac{25 (- \sqrt{3})}{2}$

Step 16.6

Multiply $- 1$ by $25$ .

$\frac{50 π}{3} + \frac{- 25 \sqrt{3}}{2}$

Step 16.7

Move the negative in front of the fraction.

$\frac{50 π}{3} - \frac{25 \sqrt{3}}{2}$

Step 17

The result can be shown in multiple forms.

Exact Form:

$\frac{50 π}{3} - \frac{25 \sqrt{3}}{2}$

Decimal Form:

$30.70924246 \dots$

Step 18