Question

The area of the plane figure bounded by the curve r=cosφ-sinφ, is equal to k ·π. Find k

Answer 1

The value of k is equal to 2.

Step-by-step explanation:

The curve given by the polar equation r = cos(φ) - sin(φ) defines a shape in the polar coordinate system. To find the area bounded by this curve, we need to calculate the integral of
`\( (1)/(2) r^2 \)` with respect to φ over the specified interval. The integral for the area (A) can be expressed as:

`\[ A = \int_(a)^(b) (1)/(2) r^2 dφ \]`

In this case, the interval φ is typically
`[0, \( \pi \)]` since we are interested in a half rotation. Substituting the given equation for r, we get:

`\[ A = \int_(0)^(\pi) (1)/(2) (cos(φ) - sin(φ))^2 dφ \]`

Solving this integral yields the area A. To express the area in terms of
`\( k \) times \( π \), we set \( A = kπ \) and solve for \( k \)`. After performing the calculations, we find that k = 2.

Therefore, the area of the plane figure bounded by the curve is equal to 2π. The key step is to correctly set up and solve the integral using the given polar equation, ensuring that the limits of integration match the desired portion of the curve.