To calculate: The area common to the regions formed by the inequalities and . Also sketch the region that satisfy both the inequalities.
The area of the region is . The graph of the region is provided below,
The inequalities and .
The standard form of the equation of the circle is , where denote the center of the circle and r denote the radius.
Consider the provided inequalities and .
Rewrite the inequality as an equation of circle, .
Recall that the standard form of the equation of the circle is , where denote the center of the circle and r denote the radius.
Compare, and also radius r is 3.
Therefore, inequality is a circle with center at and radius 3.
Now, the inequality contains all point in the coordinate plane that satisfy .
Also, is . That is plot the equation of lines and . Shade the region that lie between the lines.
Red shaded region represents and the blue shaded region represents
Now, the region common to the inequalities is shaded by green color which is a sector with included angle as and radius as 3 units.
The area of circular sector is expressed as .
Substitute r as 3 and in the above expression,
Thus, the area of the region is .
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