Find the area of the region lying in the first quadrant and bounded by y = 4 x 2 , x = 0, y = 1 and y = 4
We have to find the area enclosed by the parabola whose equation is y = 4 x 2 , x = 0 , y = 1 and y = 4 in the first quadrant. So, draw the graphs of the equations and shade the common region.
Figure (1)
To find the area bounded by the region ABCDA, assume a horizontal strip of infinitely small width and integrate the area.
Area of the region ABCDA = ∫ 1 4 x d y
From the equation of parabola, find the value of x in terms of y and substitute in the above integral.
Area of the region ABCDA = ∫ 1 4 y 2 d y = 1 2 ∫ 1 4 y d y = 1 2 [ y 1 2 + 1 1 2 + 1 ] 1 4 = 1 2 ⋅ 2 3 [ y 3 2 ] 1 4
Simplify further,
Area of the region ABCDA = 1 3 [ ( 4 ) 3 2 − ( 1 ) 3 2 ] = 1 3 [ 8 − 1 ] = 7 3 sq units
Thus, the area of required region is 7 3 sq units .
We have to find the area enclosed by the parabola whose equation is
Figure (1)
To find the area bounded by the region ABCDA, assume a horizontal strip of infinitely small width and integrate the area.
From the equation of parabola, find the value of x in terms of y and substitute in the above integral.
Simplify further,
Thus, the area of required region is
