Question

The area between x = y 2 and x = 4 is divided into two equal parts by the line x = a , find the value of a .

Answer 1

The given equation of the curve is x= y 2 and the equation of the line is x=4. The area bounded by the parabola and line x=4 is equally divided into two parts by the straight line x=a. Draw the graph of the parabola and the straight lines and label the respective points.

Figure (1)

Area of the region ADOA is equal to the area of the region ABCD.

From the above figure, it can be observed that the curve and regions are symmetric about x-axis.

Area of the region OEDO is equal to the area of the region EFCD.

To calculate the area of the region OEDO, we take a vertical strip in the region with infinitely small width, as shown in the figure above.

To find the area of the region OEDO, integrate the area of the strip.

Area of the region OEDO= ∫ 0 a ydx (1)

The equation of the parabola is x= y 2 . From this equation find the value of y in terms of x and substitute in equation (1).

x= y 2 y= x

Substitute x for y in equation (1).

Area of the region OEDO= ∫ 0 a x dx = ∫ 0 a x 1 2 +1 1 2 +1 dx = 2 3 [ x 3 2 ] 0 a = 2 3 [ a 3 2 ]

To calculate the area of the region EFCDE, we take a vertical strip in the region with infinitely small width, as shown in the figure above.

To find the area of the region EFCDE, integrate the area of the strip.

Area of the region EFCDE= ∫ a 4 ydx (2)

The equation of the parabola is x= y 2 . From this equation find the value of y in terms of x and substitute in equation (1).

x= y 2 y= x

Substitute x for y in equation (2).

Area of the region EFCDE= ∫ a 4 x dx = ∫ a 4 x 1 2 +1 1 2 +1 dx = 2 3 [ x 3 2 ] a 4 = 2 3 [ 4 3 2 − a 3 2 ] sq.units

Since both the areas are equal, equate the calculated areas of the region OEDO and EFCDE.

2 3 [ a 3 2 ]= 2 3 [ 4 3 2 − a 3 2 ] [ a 3 2 ]=[ 8− a 3 2 ] 2 a 3 2 =8 a= ( 4 ) 2 3

Thus, the value of a is ( 4 ) 2 3 .