Make a rough sketch of the graph of the function y = 4 – x2, 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 – x2, 0 x 2 ...... (4)


y = – (x2 – 4) x2 = – (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: -


5.PNG


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 – x2 )


(as x0 = 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.


6
1