The ellipse and the parabola: Let R be the region bounded by the upper half of the ellipse x²/2 + y² = 1 and the parabola y = x²/√2
a. Find the area of R

검증된 단계별 안내

Identify the curves that bound the region R. The upper half of the ellipse is given by \(\frac{x^{2}}{2} + y^{2} = 1\), which can be rewritten to express \(y\) as \(y = \sqrt{1 - \frac{x^{2}}{2}}\). The parabola is given by \(y = \frac{x^{2}}{\sqrt{2}}\).

Find the points of intersection between the ellipse and the parabola by setting their \(y\)-values equal: \(\sqrt{1 - \frac{x^{2}}{2}} = \frac{x^{2}}{\sqrt{2}}\). Square both sides to eliminate the square root and solve for \(x\).

Determine the limits of integration from the intersection points found in step 2. These \(x\)-values will serve as the bounds for the integral representing the area of region R.

Set up the integral for the area of region R as the integral of the difference between the upper curve (ellipse) and the lower curve (parabola): \(\text{Area} = \int_{a}^{b} \left( \sqrt{1 - \frac{x^{2}}{2}} - \frac{x^{2}}{\sqrt{2}} \right) \, dx\), where \(a\) and \(b\) are the intersection points.

Evaluate the integral to find the area. This may involve substitution or numerical methods if the integral is not straightforward. Remember, the integral represents the area between the two curves over the interval \([a, b]\).

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주요 개념

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Equations of Conic Sections

Understanding the standard forms of conic sections like ellipses and parabolas is essential. The ellipse here is given by x²/2 + y² = 1, representing a stretched circle, while the parabola y = x²/√2 is a quadratic curve. Recognizing these forms helps in setting up the problem and identifying the region bounded by these curves.

Finding Points of Intersection

To determine the bounded region, it is crucial to find where the ellipse and parabola intersect. This involves solving the system of equations simultaneously, which provides the limits of integration for calculating the area. Accurate intersection points ensure the correct boundaries for the integral.

Definite Integration for Area Calculation

Calculating the area between curves requires setting up a definite integral with proper limits. The area of region R is found by integrating the difference between the upper curve (ellipse) and the lower curve (parabola) over the interval defined by their intersection points. This technique is fundamental in finding areas bounded by curves.

The ellipse and the parabola: Let R be the region bounded by the upper half of the ellipse x²/2 + y² = 1 and the parabola y = x²/√2a. Find the area of R