Sketch the graph of in [0,4] and determine the area of the region enclosed by the curve, the x - axis and the lines x = 0, x = 4

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.

Required area

= 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)

(as )

Substitute

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.

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