**Problem Statement:**Find the \( y \)-coordinate of the centroid of the sector in the figure below. Assume that \( R = \sqrt{34} \) and \( \theta = \frac{\pi}{4} \).\( \bar{y} = \) [ ]**Diagram Explanation:**The diagram shows a sector of a circle with:- The origin \((0,0)\) as its center.- A given angle \( \theta \).- A radius \( R \).The sector is bounded by two radii forming an angle \( \theta = \frac{\pi}{4} \) and an arc forming the top boundary. The coordinates of the centroid are marked as \((0, \bar{y})\).**Axes:**- The x-axis is labeled and extends horizontally.- The y-axis is vertical.- The endpoint of the sector on the x-axis is marked at \( R \).**Coloring:**- The sector area is shaded in beige.- The arc above the sector is outlined in blue.

according to given data , we have π4<θ<3π4 and 0<r<34 first of all , we will find the area A=∫π43π4∫034rdrdθ integrating A=∫π43π4r22034dθ =∫π43π4342-0dθ =17θπ43π4 =173π4-π4 =17×π2 =17π2now, to find the value of y-coordinate, we will use the formula y¯=1A∫π43π4∫034r2sinθdrdθ substituting the value of area and integrating y¯=217π∫π43π4r33034sinθdθ =217π∫π43π43433-0sinθdθ =217π×34343∫π43π4sinθdθ =4343π-cosθπ43π4 =4343π-cos3π4+cosπ4 =4343π12+12 =4343π×2 =8173πanswer value of y-coordinate is 8173π