Question

Find the area under the given curves and given lines: (i) y = x 2 , x = 1, x = 2 and x -axis (ii) y = x 4 , x = 1, x = 5 and x –axis

Answer 1

(i)

We have to find the area enclosed by the parabola whose equation is y= x 2 , lines x=1,x=2 and x-axis. Draw the graphs of the parabola, lines and shade the common region.

Figure (1)

To find the area bound by the parabola with x-axis, assume a vertical strip of infinitesimally small width and integrate its area.

Area of the region ADCBA= ∫ 1 2 y dx

From the equation of parabola, find the value of y in terms of x and substitute in the above integral.

Area of the region ADCBA= ∫ 1 2 x 2 dx = [ x 3 3 ] 1 2 = 1 3 [ ( 2 ) 3 − ( 1 ) 3 ] = 7 3

Thus, the area enclosed by the parabola whose equation is y= x 2 , lines x=1,x=2 and x-axis is 7 3  sq units .

(ii)

We have to find the area enclosed by the curve whose equation is y= x 4 , lines x=1,x=5 and x-axis. Draw the graphs of the parabola, lines and shade the common region.

Figure (1)

To find the area bound by the curve with x-axis, assume a vertical strip of infinitesimally small width and integrate its area.

Area of the region ADCBA= ∫ 1 5 y dx

From the equation of curve, find the value of y in terms of x and substitute in the above integral.

Area of the region ADCBA= ∫ 1 5 x 4 dx = [ x 5 5 ] 1 5 =[ ( 5 ) 5 5 − ( 1 ) 5 5 ] =625− 1 5

Further, solve the above equation.

Area of the region ADCBA=625−0.2 =624.8

Thus, the area enclosed by the parabola whose equation is y= x 4 , lines x=1,x=5 and x-axis is 624.8 sq units .