Question

Using integration find the area of the region bounded by the tangent to the curve 4y=x^2 at the point (2,1) and the lines whose equations are x=2y and x=3y-3

Answer 1

The equation of the curve is:

y = x² / 4

From point (2,1) and the equations of the two lines, we can get the limits of integration.

We substitute the value of y in each of the two equations to get the limits of integration for the values of x.

x = 2y at y=1 this equals to : x =2 - upper limit

x = 3y - 3 at y =1 this equals to : x=0 lower limit.

Integrating x² / 4 between 0 and 2 we have :

1/4 { x³ /3} from x = 0 to x=2

Substituting in the integral we get :

1/4 × 8/3 = 2/3

Area = 2/3 square units.