The area of the region enclosed by the parabola (y - 2)² = x - 1, the line x - 2y + 4 = 0 and the po

Q: Why can't we just integrate with respect to $x$?

You absolutely can, but it is a trap! If you integrate with respect to $x$, you have to split the region into two parts because the lower boundary changes from the $x$-axis to the line. Integrating with respect to $y$ keeps the boundary function consistent, saving you time and reducing the chance of calculation errors.

Q: How do I know which area to subtract?

Always sketch the graph! By plotting the curves, you see that the integral of the parabola gives the area between the curve and the $y$-axis. The line cuts through this area, creating a triangle that is not part of the region we want. Subtracting that triangle is the geometric 'cleanup' step.

Q: Why is the intersection point $(2, 3)$ so important?

The intersection point defines the upper limit of your integration. Without it, you wouldn't know where the region ends. It is the anchor point that connects the parabola and the line, allowing you to define the exact bounds of the area.

Question

The area of the region enclosed by the parabola $(y - 2)^2 = x - 1$, the line $x - 2y + 4 = 0$ and the positive coordinate axes is_________.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing on a coordinate plane, looking at a beautiful, sweeping parabola defined by $(y - 2)^2 = x - 1$. Beside it, a straight line $x - 2y + 4 = 0$ cuts across the landscape.

Your mission is to find the area trapped between these two curves and the positive coordinate axes. This is not just a calculation; it is a story of choosing the right path.

### The Intersection: Finding Our Anchor

Before we dive into the calculus, we must know where our boundaries meet. We have the parabola $(y - 2)^2 = x - 1$ and the line $x = 2y - 4$.

By substituting the line's expression for $x$ into the parabola, we get:

$$ (y - 2)^2 + 1 = 2y - 4 $$

Expanding this, we find $y^2 - 4y + 4 + 1 = 2y - 4$, which simplifies to $y^2 - 6y + 9 = 0$. This is a perfect square: $(y - 3)^2 = 0$.

Thus, our curves intersect at $y = 3$. Plugging this back into our line equation, we find $x = 2(3) - 4 = 2$. Our anchor point is $(2, 3)$.

### The Strategy: Why $dy$ is Superior

Many students instinctively reach for $dx$ integration. But look at the region. If you integrate with respect to $x$, the lower boundary changes from the $x$-axis to the line, requiring two separate integrals.

Instead, let us look at the $y$-axis. The right boundary is always the parabola, and the left boundary is the $y$-axis. By integrating with respect to $y$ from $0$ to $3$, we simplify the problem into one elegant integral.

### The Calculation: The Parabola and the Triangle

We calculate the total area under the parabola from $y = 0$ to $y = 3$ using the integral:

$$ \int_{0}^{3} ((y - 2)^2 + 1) \, dy $$

The antiderivative is:

$$ \frac{(y - 2)^3}{3} + y $$

Evaluating this from $0$ to $3$, we get:

$$ \left( \frac{1}{3} + 3 ight) - \left( -\frac{8}{3} ight) = 6 $$

This is the area between the parabola and the $y$-axis. However, our region is bounded by the line, not the $y$-axis.

We must subtract the triangular area formed by the line and the $y$-axis. The vertices are $(0, 2)$, $(2, 3)$, and $(0, 3)$.

The base is $1$ and the height is $2$, so the area is:

$$ \frac{1}{2} 	imes 1 	imes 2 = 1 $$

Finally, we subtract this triangle from our total: $6 - 1 = 5$.

The area is **$5$ square units**. You have successfully navigated the geometry, mastered the intersection, and utilized the power of integration to find the truth.

Animated Solution for Mathematics - Definite Integration: The area of the region enclosed by the parabola $(y - 2)^{2} = x - 1$ , the line $x - 2 y + 4 = 0$ and the positive coordinate axes is_________.

Enter Numerical Value:

Visualized Solution

Visualize the Curves

Find Intersection Points

Solve for $y$

Find the $x$ Coordinate

Identify the Bounded Region

Set up the Integral

Integrate the Parabola

Evaluate Parabola Integral

Calculate Triangle Area

Final Area Calculation

The Sigma Insight: Area Bounded by Curves

Analyzing the Setup

The Intersection

The Strategy

The Calculation

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