Calculus Examples

$\int_{0}^{9} \sqrt{81 - x^{2}} d x$

Step 1

Let $x = 9 \sin (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = 9 \cos (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $9 \cos (t)$ is positive.

$\int_{0}^{\frac{π}{2}} \sqrt{81 - {(9 \sin (t))}^{2}} (9 \cos (t)) d t$

Step 2

Simplify terms.

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Step 2.1

Simplify $\sqrt{81 - {(9 \sin (t))}^{2}}$ .

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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 $9 \sin (t)$ .

$\int_{0}^{\frac{π}{2}} \sqrt{81 - (9^{2} \sin^{2} (t))} (9 \cos (t)) d t$

Step 2.1.1.2

Raise $9$ to the power of $2$ .

$\int_{0}^{\frac{π}{2}} \sqrt{81 - (81 \sin^{2} (t))} (9 \cos (t)) d t$

Step 2.1.1.3

Multiply $81$ by $- 1$ .

$\int_{0}^{\frac{π}{2}} \sqrt{81 - 81 \sin^{2} (t)} (9 \cos (t)) d t$

Step 2.1.2

Factor $81$ out of $81$ .

$\int_{0}^{\frac{π}{2}} \sqrt{81 (1) - 81 \sin^{2} (t)} (9 \cos (t)) d t$

Step 2.1.3

Factor $81$ out of $- 81 \sin^{2} (t)$ .

$\int_{0}^{\frac{π}{2}} \sqrt{81 (1) + 81 (- \sin^{2} (t))} (9 \cos (t)) d t$

Step 2.1.4

Factor $81$ out of $81 (1) + 81 (- \sin^{2} (t))$ .

$\int_{0}^{\frac{π}{2}} \sqrt{81 (1 - \sin^{2} (t))} (9 \cos (t)) d t$

Step 2.1.5

Apply pythagorean identity.

$\int_{0}^{\frac{π}{2}} \sqrt{81 \cos^{2} (t)} (9 \cos (t)) d t$

Step 2.1.6

Rewrite $81 \cos^{2} (t)$ as ${(9 \cos (t))}^{2}$ .

$\int_{0}^{\frac{π}{2}} \sqrt{{(9 \cos (t))}^{2}} (9 \cos (t)) d t$

Step 2.1.7

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

$\int_{0}^{\frac{π}{2}} 9 \cos (t) (9 \cos (t)) d t$

Step 2.2

Simplify.

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Step 2.2.1

Multiply $9$ by $9$ .

$\int_{0}^{\frac{π}{2}} 81 \cos (t) \cos (t) d t$

Step 2.2.2

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

$\int_{0}^{\frac{π}{2}} 81 (\cos^{1} (t) \cos (t)) d t$

Step 2.2.3

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

$\int_{0}^{\frac{π}{2}} 81 (\cos^{1} (t) \cos^{1} (t)) d t$

Step 2.2.4

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

$\int_{0}^{\frac{π}{2}} 81 {\cos (t)}^{1 + 1} d t$

Step 2.2.5

Add $1$ and $1$ .

$\int_{0}^{\frac{π}{2}} 81 \cos^{2} (t) d t$

Step 3

Since $81$ is constant with respect to $t$ , move $81$ out of the integral.

$81 \int_{0}^{\frac{π}{2}} \cos^{2} (t) d t$

Step 4

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

$81 \int_{0}^{\frac{π}{2}} \frac{1 + \cos (2 t)}{2} d t$

Step 5

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

$81 (\frac{1}{2} \int_{0}^{\frac{π}{2}} 1 + \cos (2 t) d t)$

Step 6

Combine $\frac{1}{2}$ and $81$ .

$\frac{81}{2} \int_{0}^{\frac{π}{2}} 1 + \cos (2 t) d t$

Step 7

Split the single integral into multiple integrals.

$\frac{81}{2} (\int_{0}^{\frac{π}{2}} d t + \int_{0}^{\frac{π}{2}} \cos (2 t) d t)$

Step 8

Apply the constant rule.

$\frac{81}{2} (t]_{0}^{\frac{π}{2}} + \int_{0}^{\frac{π}{2}} \cos (2 t) d t)$

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 $u = 2 t$ .

$u_{lower} = 2 \cdot 0$

Step 9.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

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.

$u_{lower} = 0$

$u_{upper} = π$

Step 9.7

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

$\frac{81}{2} (t]_{0}^{\frac{π}{2}} + \int_{0}^{π} \cos (u) \frac{1}{2} d u)$

Step 10

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

$\frac{81}{2} (t]_{0}^{\frac{π}{2}} + \int_{0}^{π} \frac{\cos (u)}{2} d u)$

Step 11

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

$\frac{81}{2} (t]_{0}^{\frac{π}{2}} + \frac{1}{2} \int_{0}^{π} \cos (u) d u)$

Step 12

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

$\frac{81}{2} (t]_{0}^{\frac{π}{2}} + \frac{1}{2} \sin (u)]_{0}^{π})$

Step 13

Substitute and simplify.

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Step 13.1

Evaluate $t$ at $\frac{π}{2}$ and at $0$ .

$\frac{81}{2} ((\frac{π}{2}) + 0 + \frac{1}{2} \sin (u)]_{0}^{π})$

Step 13.2

Evaluate $\sin (u)$ at $π$ and at $0$ .

$\frac{81}{2} ((\frac{π}{2}) + 0 + \frac{1}{2} ((\sin (π)) - \sin (0)))$

Step 13.3

Add $\frac{π}{2}$ and $0$ .

$\frac{81}{2} (\frac{π}{2} + \frac{1}{2} (\sin (π) - \sin (0)))$

Step 14

Simplify.

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Step 14.1

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

$\frac{81}{2} (\frac{π}{2} + \frac{1}{2} (\sin (π) - 0))$

Step 14.2

Multiply $- 1$ by $0$ .

$\frac{81}{2} (\frac{π}{2} + \frac{1}{2} (\sin (π) + 0))$

Step 14.3

Add $\sin (π)$ and $0$ .

$\frac{81}{2} (\frac{π}{2} + \frac{1}{2} \sin (π))$

Step 14.4

Combine $\frac{1}{2}$ and $\sin (π)$ .

$\frac{81}{2} (\frac{π}{2} + \frac{\sin (π)}{2})$

Step 15

Simplify.

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Step 15.1

Combine the numerators over the common denominator.

$\frac{81}{2} \cdot \frac{π + \sin (π)}{2}$

Step 15.2

Simplify each term.

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Step 15.2.1

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

$\frac{81}{2} \cdot \frac{π + \sin (0)}{2}$

Step 15.2.2

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

$\frac{81}{2} \cdot \frac{π + 0}{2}$

Step 15.3

Add $π$ and $0$ .

$\frac{81}{2} \cdot \frac{π}{2}$

Step 15.4

Multiply $\frac{81}{2} \cdot \frac{π}{2}$ .

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Step 15.4.1

Multiply $\frac{81}{2}$ by $\frac{π}{2}$ .

$\frac{81 π}{2 \cdot 2}$

Step 15.4.2

Multiply $2$ by $2$ .

$\frac{81 π}{4}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$\frac{81 π}{4}$

Decimal Form:

$63.61725123 \dots$

Step 17