Calculus Examples

$\int_{- 2}^{2} \sqrt{2^{2} - x^{2}} d x$

Step 1

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

$\int_{- 2}^{2} \sqrt{4 - x^{2}} d x$

Step 2

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

$\int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{4 - {(2 \sin (t))}^{2}} (2 \cos (t)) d t$

Step 3

Simplify terms.

Tap for more steps...

Step 3.1

Simplify $\sqrt{4 - {(2 \sin (t))}^{2}}$ .

Tap for more steps...

Step 3.1.1

Simplify each term.

Tap for more steps...

Step 3.1.1.1

Apply the product rule to $2 \sin (t)$ .

$\int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{4 - (2^{2} \sin^{2} (t))} (2 \cos (t)) d t$

Step 3.1.1.2

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

$\int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{4 - (4 \sin^{2} (t))} (2 \cos (t)) d t$

Step 3.1.1.3

Multiply $4$ by $- 1$ .

$\int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{4 - 4 \sin^{2} (t)} (2 \cos (t)) d t$

Step 3.1.2

Factor $4$ out of $4$ .

$\int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{4 (1) - 4 \sin^{2} (t)} (2 \cos (t)) d t$

Step 3.1.3

Factor $4$ out of $- 4 \sin^{2} (t)$ .

$\int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{4 (1) + 4 (- \sin^{2} (t))} (2 \cos (t)) d t$

Step 3.1.4

Factor $4$ out of $4 (1) + 4 (- \sin^{2} (t))$ .

$\int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{4 (1 - \sin^{2} (t))} (2 \cos (t)) d t$

Step 3.1.5

Apply pythagorean identity.

$\int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{4 \cos^{2} (t)} (2 \cos (t)) d t$

Step 3.1.6

Rewrite $4 \cos^{2} (t)$ as ${(2 \cos (t))}^{2}$ .

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

Step 3.1.7

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

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

Step 3.2

Simplify.

Tap for more steps...

Step 3.2.1

Multiply $2$ by $2$ .

$\int_{- \frac{π}{2}}^{\frac{π}{2}} 4 \cos (t) \cos (t) d t$

Step 3.2.2

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

$\int_{- \frac{π}{2}}^{\frac{π}{2}} 4 (\cos^{1} (t) \cos (t)) d t$

Step 3.2.3

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

$\int_{- \frac{π}{2}}^{\frac{π}{2}} 4 (\cos^{1} (t) \cos^{1} (t)) d t$

Step 3.2.4

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

$\int_{- \frac{π}{2}}^{\frac{π}{2}} 4 {\cos (t)}^{1 + 1} d t$

Step 3.2.5

Add $1$ and $1$ .

$\int_{- \frac{π}{2}}^{\frac{π}{2}} 4 \cos^{2} (t) d t$

Step 4

Since $4$ is constant with respect to $t$ , move $4$ out of the integral.

$4 \int_{- \frac{π}{2}}^{\frac{π}{2}} \cos^{2} (t) d t$

Step 5

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

$4 \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{1 + \cos (2 t)}{2} d t$

Step 6

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

$4 (\frac{1}{2} \int_{- \frac{π}{2}}^{\frac{π}{2}} 1 + \cos (2 t) d t)$

Step 7

Simplify.

Tap for more steps...

Step 7.1

Combine $\frac{1}{2}$ and $4$ .

$\frac{4}{2} \int_{- \frac{π}{2}}^{\frac{π}{2}} 1 + \cos (2 t) d t$

Step 7.2

Cancel the common factor of $4$ and $2$ .

Tap for more steps...

Step 7.2.1

Factor $2$ out of $4$ .

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

Step 7.2.2

Cancel the common factors.

Tap for more steps...

Step 7.2.2.1

Factor $2$ out of $2$ .

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

Step 7.2.2.2

Cancel the common factor.

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

Step 7.2.2.3

Rewrite the expression.

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

Step 7.2.2.4

Divide $2$ by $1$ .

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

Step 8

Split the single integral into multiple integrals.

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

Step 9

Apply the constant rule.

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

Step 10

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

Tap for more steps...

Step 10.1

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

Tap for more steps...

Step 10.1.1

Differentiate $2 t$ .

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

Step 10.1.2

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 10.1.3

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 10.1.4

Multiply $2$ by $1$ .

$2$

Step 10.2

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

$u_{lower} = 2 (- \frac{π}{2})$

Step 10.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 10.3.1

Move the leading negative in $- \frac{π}{2}$ into the numerator.

$u_{lower} = 2 \frac{- π}{2}$

Step 10.3.2

Cancel the common factor.

$u_{lower} = 2 \frac{- π}{2}$

Step 10.3.3

Rewrite the expression.

$u_{lower} = - π$

Step 10.4

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

$u_{upper} = 2 \frac{π}{2}$

Step 10.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 10.5.1

Cancel the common factor.

$u_{upper} = 2 \frac{π}{2}$

Step 10.5.2

Rewrite the expression.

$u_{upper} = π$

Step 10.6

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

$u_{lower} = - π$

$u_{upper} = π$

Step 10.7

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

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

Step 11

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

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

Step 12

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

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

Step 13

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

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

Step 14

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

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

Step 15

Substitute and simplify.

Tap for more steps...

Step 15.1

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

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

Step 15.2

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

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

Step 15.3

Simplify.

Tap for more steps...

Step 15.3.1

Combine the numerators over the common denominator.

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

Step 15.3.2

Add $π$ and $π$ .

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

Step 15.3.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 15.3.3.1

Cancel the common factor.

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

Step 15.3.3.2

Divide $π$ by $1$ .

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

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

Step 16

Simplify.

Tap for more steps...

Step 16.1

Simplify the numerator.

Tap for more steps...

Step 16.1.1

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

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

Step 16.1.2

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

$2 (π + \frac{0 - \sin (- π)}{2})$

Step 16.1.3

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

$2 (π + \frac{0 - \sin (0)}{2})$

Step 16.1.4

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

$2 (π + \frac{0 - 0}{2})$

Step 16.1.5

Multiply $- 1$ by $0$ .

$2 (π + \frac{0 + 0}{2})$

Step 16.1.6

Add $0$ and $0$ .

$2 (π + \frac{0}{2})$

Step 16.2

Divide $0$ by $2$ .

$2 (π + 0)$

Step 17

Add $π$ and $0$ .

$2 π$

Step 18

The result can be shown in multiple forms.

Exact Form:

$2 π$

Decimal Form:

$6.28318530 \dots$

Step 19