Question

Draw a rough sketch to indicate the bounded between the curve y² = 4x and the line x = 3. Also, find the area of this region

Answer 1

Given equations are:

x = 3 ...... (1)

And y² = 4x ...... (2)

Equation (1) represents a line parallel to the y - axis at a distance of 3 units and equation (2) represents a parabola with vertex at origin and x - axis as its axis; A rough sketch is given as below:

We have to find the area of shaded region.

Required area

= shaded region OBCAO

= 2 (shaded region OBCO) (as it is symmetrical about the x - axis)

(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)

\(=2\int^3_0 y\,dx\) (As x is between (0,3) and the value of y varies)

On integrating we get,

On applying the limits, we get,

Hence the area of the region bounded between the line x = 3 and the parabola y² = 4x is equal to \(8\sqrt{3}\) square units.