The area bounded by the parabola x = 4 - y² and y-axis, in square units, is

Concept:

Area under a Curve by Integration

Find the area under this curve by summing vertically.

• In this case, we find the area is the sum of the rectangles, height x = f(y) and width dy.
• If we are given y = f(x), then we need to re-express this as x = f(y) and we need to sum from the bottom to top.

So, \({\bf{A}} = \mathop \smallint \nolimits_{\bf{a}}^{\bf{b}} {\bf{xdy}} = \mathop \smallint \nolimits_{\bf{a}}^{\bf{b}} {\bf{f}}\left( {\bf{y}} \right){\bf{dy}}\)

Calculation:

Given Curve: x = 4 - y2

⇒ y2 = 4 - x
⇒ y2 = - (x - 4)           

The above curve is the equation of the Parabola,

We know that at y-axis; x = 0

⇒ y2 = 4 - x

⇒ y2 = 4 - 0 = 4

⇒ y = ± 2

⇒ (0, 2) or (0, -2) are Point of intersection.

Area under the curve \( = \mathop \int \nolimits_{-2}^2 {\rm{xdy}}\)

\(= \rm \int_{-2}^2 (4-y^2)dy\)

\(\rm = \left[4y- {\frac{{{{ {{\rm{y}} } }^3}}}{3}} \right]_{-2}^2\)

\(= \frac{32}{3}\) Sq. unit