Definite Integrals, Area Bounded by the Curve x = f (y) Between Y-Axis and the Lines y = c, y = d

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Area Bounded by the Curve X=F(y) between Y-Axis and the Lines Y=C,Y=D:

  • Let be the curve and let be the abscissae at respectively.

  • Let be any point on the curve and let be a neighbouring point on it.

  • Draw and perpendiculars on y-axis from and respectively. As changes, the area () also changes and hence clearly a function of.

  • Let denote the area (), then the area () will be.

  • The area () Area () Area ().

Area Bounded by the Curve x=f(y) between y-axis and the Lines y=c,y=d

Area Bounded by the Curve X=F(y) between Y-Axis

Area Bounded by the Curve x=f(y) between y-axis and the Lines y=c,y=d

Complete the rectangle. Then the area () lies between the area () and the area (), that is,

lies between and

lies between and

In the limiting position when and

lies between and

Integrating both sides with respect to, between the limits to , we get

(Area when)(Area when)

Area ()

Area ()

Hence area

The area bounded by the curve, the y-axis and the lines and is

or

Where is a continuous single valued function and does not change sign in the interval .

Example:

Find the area bounded by the curve, y-axis and the lines.

Solution:

The given curve is .

Required area bounded by the curve, y-axis and the lines is

The curve of y=x, y=3

The Curve of Y=X, Y=3

The curve of y=x, y=3

square units

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