Question

Find the area of the region. Interior of r = 2 + cos (theta)

Answer 1

The area of the region, the interior of
`\( r = 2 + \cos(\theta) \)`, is
`\( (3\pi)/(2) \)` square units.

Step-by-step explanation:

To find the area of the region enclosed by the polar curve
`\( r = 2 + \cos(\theta) \)`, we can use the polar area formula
`\( A = (1)/(2) \int_(\alpha)^(\beta) [f(\theta)]^2 \, d\theta \)`, where
`\( f(\theta) \)`represents the radius function.

In this case,
`\( f(\theta) = 2 + \cos(\theta) \).` The limits of integration
`(\( \alpha \)` and
`\( \beta \))`should be determined by finding where the curve intersects the pole, which occurs when
`\( r = 0 \)`. Solving
`\( 2 + \cos(\theta) = 0 \)` gives . Thus, the limits are
`\( \alpha = 0 \)` and
`\( \beta = \pi \)`.

Now, substitute these values into the polar area formula:

`\[ A = (1)/(2) \int_(0)^(\pi) [2 + \cos(\theta)]^2 \, d\theta \]`

The integration may involve trigonometric functions, and simplifying the expression inside the integral and evaluating the definite integral will yield the area. The result is
`\( (3\pi)/(2) \)` square units.

Finding the area of a region enclosed by a polar cur
`\( \theta = \pi \)`ve involves converting the polar coordinates to rectangular coordinates and applying standard calculus techniques.