Introduction

The problem of finding the area bounded by the curve ( x y^2 - 2y ) and the y-axis is a classic example in integral calculus that requires a solid understanding of integration techniques and geometric interpretations. This article will guide you through the detailed steps to calculate this area, ensuring that you fully grasp the concepts involved.

Intersection with the Y-Axis

To begin, we need to determine the points where the curve x y^2 - 2y intersects the y-axis. This occurs when x 0. Let's solve for y:

(0 y^2 - 2y)

Factoring the equation, we get:

(y(y - 2) 0)

Thus, the solutions are:

y 0 and y 2

Setting Up the Integral

The area A bounded by the curve and the y-axis from y 0 to y 2 can be found by solving the integral:

A ∫y?y? x dy

Substituting x y^2 - 2y, we have:

A ∫02 (y^2 - 2y) dy

Calculating the Integral

Let's calculate the integral step by step:

A ∫02 (y^2 - 2y) dy left [ frac{y^3}{3} - y^2 right ]02

Evaluating at the bounds:

left ( frac{2^3}{3} - 2^2 right ) - left ( frac{0^3}{3} - 0^2 right )

(frac{8}{3} - 4 - 0 frac{8}{3} - frac{12}{3} -frac{4}{3})

Since area cannot be negative, we take the absolute value:

A (frac{4}{3})

Conclusion

The area bounded by the curve ( x y^2 - 2y ) and the y-axis is:

(boxed{frac{4}{3}})

Algebraic and Geometric Interpretation

It's important to note that the algebraic area calculated can be negative, which indicates the enclosed area is on the negative x-axis side. In this case, the actual area is positive and should be taken as:

(boxed{frac{4}{3}})

Alternative Perspective

If we consider the curve ( y x^2 - 2x ), finding the area bounded by this curve and the x-axis can be approached similarly but using the integral with respect to x instead:

F(x) (frac{x^3}{3} - x^2)

Then, calculating the area:

A F(2) - F(0) (8/3 - 4) - 0 8/3 - 12/3 -4/3

The negative sign here also indicates the enclosed area is below the x-axis, so the actual positive area is:

(boxed{frac{4}{3}})