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 defined by the equation \( (x-7)^{2}+(y+5)^{2}=25 \) using integration, we first recognize that this represents a circle with a center at \( (7, -5) \) and a radius of \( 5 \), since \( \sqrt{25} = 5 \). The area \( A \) of a circle can be found by integrating the area under the curve defined by the circle. By rearranging for \( y \), we can express it as \( y = -5 + \sqrt{25 - (x-7)^{2}} \) for the upper half and \( y = -5 - \sqrt{25 - (x-7)^{2}} \) for the lower half. The area can be computed using the integral: \[ A = 2 \int_{2}^{12} \left(-5 + \sqrt{25 - (x-7)^{2}}\right) \, dx \] Calculating the definite integral from \( x=2 \) to \( x=12 \) yields the area of the circle, but for simplicity, we can also apply the formula \( A = \pi r^2 = \pi (5)^2 = 25\pi \), which is approximately \( 78.54 \) square units. Now let’s not forget that the full circle, regardless of the method, has an area calculated simply by the radius squared multiplied by \( \pi \)!