Make a rough sketch of the graph of the function y = 4 – x^{2}, 0 ≤ x ≤ 2 and determine the area enclosed by the curve, the x - axis and the lines x = 0 and x = 2.

Given equations are:

x – axis ...... (1)

x = 0 ...... (2)

x = 2 ...... (3)

And y = 4 – x^{2}, 0 ≤ x ≤ 2 ...... (4)

⇒ y = – (x^{2} – 4) ⇒ x^{2} = – (y – 4)

equation (4) represents a downward parabola with vertex at (0,4) and passing through (2,0) and ( – 2,0) on x – axis, equation (3) represents a line parallel to y – axis at a distance of 2 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 OABO

(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 (0,2) and the value of y varies)

(as y = 4 – x^{2} )

(as x^{0} = 1)

On integrating we get,

On applying the limits, we get,

Hence the area enclosed by the curve, the x - axis and the lines x = 0 and x = 2 is equal to square units.

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