 Hello and welcome to the session. I am Deepika here. Let's discuss the question which says Choose the correct answer in the following Area lying in the first quadrant and bounded by the circle x square plus y square is equal to 4 and the lines x is equal to 0 and x is equal to 2 is a pi b pi by 2 c pi by 3 and t pi by 4 Now we know that the area bounded by the curve y is equal to fx and the lines x is equal to 0 and x is equal to a in the first quadrant is given by a is equal to integral from 0 to a y dx and this is again equal to integral from 0 to a f of x dx So this is a key idea behind our question. We will take the help of this key idea to solve the above questions. Let's start the solution Now first of all we will draw the figure to find the region bounded by the circle x square plus y square is equal to 4 and the lines x is equal to 0 and x is equal to 2 Now the equation x square plus y square is equal to 4 represents a circle with center at the origin and radius 2 and the line x is equal to 0 is y axis and x is equal to 2 is a line parallel to y axis So we have to find the area of this shaded region Now the shaded region the region lying in the first quadrant bounded by the circle x square plus y square is equal to 4 and the lines x is equal to 0 and x is equal to 2 By taking vertical strips of height y and width dx the area of this shaded region is equal to integral from 0 to 2 y dx Now we have plus y square is equal to 4 This implies y is equal to the first quadrant Now by using this integral we have area a is equal to x upon 2 into under root of a square minus x square Now a is 2 here so 2 square 4 minus x square plus upon 2 sin inverse x upon 2 and the limits are 0 to 2 So this is equal to 4 over 2 into under root of 4 minus 2 square 2 over 2 minus 0 over 2 into under root of 4 minus 0 plus 2 sin inverse 0 over 2 so this is equal to 0 plus 2 sin inverse 1 minus 0 plus 0 because sin 0 is 0 so sin inverse 0 is also 0 and this is equal to 2 sin inverse 1 and this is equal to now sin pi by 2 is equal to 1 so sin inverse 1 is equal to pi by 2 so 2 into pi by 2 this is equal to 5 The area lying in the first quadrant and bounded by this circle x square plus y square is equal to 4 and the lines x is equal to 0 and x is equal to 2 is 5 This is our option A. Hope the solution is clear to you. Bye and take care