 Good morning friends. I am Purva and today I will help you with the following question. Choose the correct answer in the following and the question is the area bounded by the curve y is equal to x into mod x, x-axis and the ordinate x is equal to minus 1 and x is equal to 1 is given by a 0 b 1 upon 3 c 2 upon 3 d 4 upon 3. Let us now begin with the solution. Now we have to find the area bounded by the curve y is equal to x into mod x, x-axis and the lines x is equal to minus 1 and x is equal to 1. Now y is equal to x into mod x is equivalent to two curves. The first is y is equal to x into x that is y is equal to x square if x is greater than equal to 0 and the second one is y is equal to x into minus x that is y is equal to minus x square if x is less than 0. Now for y is equal to x square we give positive values to x hence it lies above x-axis that is in the first quadrant whereas for y is equal to minus x square we have to give negative values to x hence y is also negative hence this curve lies in the third quadrant. Now y is equal to x square passes through the points 0 comma 0 1 comma 1 2 comma 4 and so on. So by giving value 0 to x we get y is equal to 0 by giving x value 1 we get y is equal to 1 and by giving x value 2 we get y is equal to 4 and so on and y is equal to minus x square passes through the points 0 comma 0 minus 1 comma minus 1 minus 2 comma minus 4 and so on. So giving value 0 to x we get y is equal to 0 giving value minus 1 to x we get y is equal to minus 1 giving value minus 2 to x we get y is equal to minus 4 and so on. And x is equal to minus 1 and x is equal to 1 are the two lines which are parallel to y-axis. So using this information we get the following figure where this is the curve y is equal to x square for x greater than equal to 0 and here below it becomes y is equal to minus x square for x less than 0 and these are the two lines x is equal to 1 and x is equal to minus 1 parallel to y-axis. And we have to find the area of this shaded region. Therefore required area is equal to now we must remember that we have to take the absolute value of the area of the region below x-axis. Now this region is bounded by the curve y is equal to minus x square and the limit is from minus 1 to 0. So we have therefore required area is equal to modulus of integral, limit is from minus 1 to 0 minus x square dx plus and this shaded region is bounded by the curve y is equal to x square for x greater than equal to 0 and we have limit from 0 to 1. So we get plus integral limit is from 0 to 1 x square dx and this is equal to modulus of now integral of minus x square is minus x cube upon 3 and limit is from minus 1 to 0 plus integral of x square is x cube upon 3 and limit is from 0 to 1 and this is equal to now putting the limits we get mod of 0 minus 1 upon 3. Putting up a limit 0 in place of x we get 0 minus putting lower limit minus 1 in place of x we get 1 upon 3 plus again putting the limits we get 1 upon 3 minus 0. Putting up a limit 1 in place of x we get 1 upon 3 minus putting lower limit 0 we get 0. This is equal to modulus of minus 1 upon 3 plus 1 upon 3 and this is further equal to now modulus of minus 1 upon 3 is 1 upon 3 plus 1 upon 3 and this is equal to 2 upon 3 therefore we get required area is equal to 2 upon 3 and this is same as part C Therefore we get the correct answer is C. Hope you have understood the solution. Bye and take care.