Given equations are:
x – axis ...... (1)
x = 0 ...... (2)
x = 4 ...... (3)
And , 0 ≤ x ≤ 4 ...... (4)
equation (4) represents a half parabola with vertex at ( – 1,0) and passing through (4,0) on x – axis, equation (3) represents a line parallel to y - axis at a distance of 4 units and equation (2) represents y - axis.
A rough sketch is given as below: -
We have to find the area of the shaded region.
= shaded region AOBC
(the area can be found by taking a small slice in each region of width Δx, then the area of that sliced part will be yΔx as it is a rectangle and then integrating it to get the area of the whole region)
(As x is between (0,4) and the value of y varies)
So the above equation becomes,
On integrating we get,
On applying the limits we get,
Hence the area of the region enclosed by the curve, the x - axis and the lines x = 0, x = 4 is equal to square units.