Calculus Examples

r = 3 sec (θ)

$r = 3 \sec (θ)$

Step 1

Simplify the right side.

Tap for more steps...

Step 1.1

Simplify

3 sec (θ)

$3 \sec (θ)$ .

Tap for more steps...

Step 1.1.1

Rewrite

sec (θ)

$\sec (θ)$ in terms of sines and cosines.

r = 3 \frac{1}{cos (θ)}

$r = 3 \frac{1}{\cos (θ)}$

Step 1.1.2

Combine

3

$3$ and

$\frac{1}{\cos (θ)}$ .

$r = \frac{3}{\cos (θ)}$

$r = \frac{3}{\cos (θ)}$

$r = \frac{3}{\cos (θ)}$

Step 2

Since

$\cos (θ) = \frac{x}{r}$ , replace

$\cos (θ)$ with

$\frac{x}{r}$ .

$r = \frac{3}{\frac{x}{r}}$

Step 3

Multiply both sides by

$r$ .

Tap for more steps...

Step 3.1

Multiply both sides by

$r$ .

$r \cdot r = r \frac{3}{\frac{x}{r}}$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Multiply

$r$ by

$r$ .

$r^{2} = r \frac{3}{\frac{x}{r}}$

$r^{2} = r \frac{3}{\frac{x}{r}}$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Simplify

$r \frac{3}{\frac{x}{r}}$ .

Tap for more steps...

Step 3.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$r^{2} = r (3 \frac{r}{x})$

Step 3.3.1.2

Rewrite using the commutative property of multiplication.

$r^{2} = 3 r \frac{r}{x}$

Step 3.3.1.3

Multiply

$3 r \frac{r}{x}$ .

Tap for more steps...

Step 3.3.1.3.1

Combine

$3$ and

$\frac{r}{x}$ .

$r^{2} = r \frac{3 r}{x}$

Step 3.3.1.3.2

Combine

$r$ and

$\frac{3 r}{x}$ .

$r^{2} = \frac{r (3 r)}{x}$

Step 3.3.1.3.3

Raise

$r$ to the power of

$1$ .

$r^{2} = \frac{3 (r^{1} r)}{x}$

Step 3.3.1.3.4

Raise

$r$ to the power of

$1$ .

$r^{2} = \frac{3 (r^{1} r^{1})}{x}$

Step 3.3.1.3.5

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$r^{2} = \frac{3 r^{1 + 1}}{x}$

Step 3.3.1.3.6

Add

$1$ and

$1$ .

$r^{2} = \frac{3 r^{2}}{x}$

$r^{2} = \frac{3 r^{2}}{x}$

$r^{2} = \frac{3 r^{2}}{x}$

$r^{2} = \frac{3 r^{2}}{x}$

$r^{2} = \frac{3 r^{2}}{x}$

Step 4

Since

$r^{2} = x^{2} + y^{2}$ , replace

$r^{2}$ with

$x^{2} + y^{2}$ and

$r$ with

$\sqrt{x^{2} + y^{2}}$ .

$x^{2} + y^{2} = \frac{3 (x^{2} + y^{2})}{x}$

$r = 3 \sec θ$

(

(

)

)

|

|

[

[

]

]

√

√





≥

≥





∫

∫













7

7

8

8

9

9













≤

≤





°

°

θ

θ









4

4

5

5

6

6

/

/

^

^

×

×

>

>





π

π













1

1

2

2

3

3

-

-

+

+

÷

÷

<

<





∞

∞





!

!



,

,

0

0

.

.

%

%



=

=









$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$