Question

Find the area of the region bounded by the curves y = x 2 + 2, y = x , x = 0 and x = 3

Answer 1

We have to find the area bounded by curve y= x 2 +2, y=x, x=0 and x=3. Draw the graphs of the equations and shade the common region.

Figure (1)

To find the area of the region OABDO, assume a vertical strip and integrate the area of that strip.

Area of the region OABDO= ∫ 0 3 ydx

From the equation of the curve y= x 2 +2, find the value of y in terms of x and put it in the above integral.

Area of the region OABDO= ∫ 0 3 ( x 2 +2 )dx = [ x 3 3 +2x ] 0 3 =[ 27 3 +6 ] =15 sq units

The area of triangle OCD is calculated as,

Area of the triangle OCD= 1 2 ×3×3 = 9 2  sq units

The equation for the area of OABCO is,

Area of the region OABCO=Area of the region OABDO−Area of the triangle OCD =15− 9 2 = 21 2  sq units

Thus, the required area bounded by the curve y= x 2 +2, y=x, x=0 and x=3 is 21 2  sq units.