Calculate the area of the circle for med by the following equation using integration \( (x-7)^{2}+(y+5)^{2}=25 \)

To calculate the area of the circle given by the equation \( (x-7)^{2}+(y+5)^{2}=25 \), we realize that this is the standard form of a circle centered at \( (7, -5) \) with a radius \( r = 5 \) (since \( r^2 = 25 \)). The area of a circle can also be confirmed using integration by setting up the following integral in terms of \( y \): \[ A = 2 \int_{-5}^{0} \sqrt{25 - (y + 5)^{2}} \, dy \] This computes half of the area because we will then multiply by 2 to account for both halves of the circle, ultimately leading to the formula \( A = \pi r^2 \), which gives \( A = 25\pi \). To apply this area practical, circles show up everywhere—in nature and in city planning! Think about how roundabouts create smoother traffic flow or how the design of sporting fields, like baseball diamonds, hinges on circular geometry. Understanding areas of circles equips you with the tools to design engaging outdoor spaces or plan effective transportation routes. Who knew mathematics could shape the world so literally?