# Area for grazing by the goat tied to a silo

# Perimeter of the curve r = 4(1 + sin theta) by integration

# Length of Arc in Polar Plane | Applications of Integration

The length of arc on polar plane is given by the formula:

$\displaystyle s = \int_{\theta_1}^{\theta_2} \sqrt{r^2 + \left( \dfrac{dr}{d\theta} \right)^2} ~ d\theta$

The formula above is derived in two ways. See it here: http://www.mathalino.com/reviewer/integral-calculus/length-arc-polar-pla...

# Length of Arc in XY-Plane | Applications of Integration

The length of arc in rectangular coordinates is given by the following formulas:

$\displaystyle s = \int_{x_1}^{x_2} \sqrt{1 + \left( \dfrac{dy}{dx} \right)^2} \, dx$ and $\displaystyle s = \int_{y_1}^{y_2} \sqrt{1 + \left(\dfrac{dx}{dy} \right)^2} \, dy$

See the derivations here: http://www.mathalino.com/reviewer/integral-calculus/length-arc-xy-plane-...

# Longitude of an airplane crossing the equator

# Rate of change of surface area of sphere

**Problem**

Gas is escaping from a spherical balloon at the rate of 2 cm^{3}/min. Find the rate at which the surface area is decreasing, in cm^{2}/min, when the radius is 8 cm..