Calculating the Area Between yx^2 and xy^2 using Symmetry and Integration
When dealing with the area between the parabolas yx^2 and xy^2, we need to understand the intersection points and determine the bounds of the region of interest. We can then use integration or symmetry to calculate the area.
Intersection Points and Graphs
The parabolas yx^2 and xy^2 intersect at (0,0) and (1,1). To visualize the area in question, draw both graphs. The curve xy^2 is above yx^2 in the interval [0,1].
Integrating the Area Using Symmetry
Instead of integrating directly, we can use symmetry to simplify the calculation. The area can be seen as the sum of the areas of two equal regions. Thus, we can find the area of one region and then multiply by 2.
Consider the integral:
[ A 2 int_0^1 (sqrt{x} - x^2) , dx ]
Let's evaluate this integral:
[ A 2 left[ frac{2}{3}x^{3/2} - frac{1}{3}x^3 right]_0^1 ]
[ A 2 left( frac{2}{3} - frac{1}{3} right) frac{2}{3} ]
Hence, the area of the region between the parabolas is ( frac{1}{3} ) units2.
Alternative Approach Using Chords and Geometry
Archimedes' method involves using chords to find the area. By bisecting the chord, we can use the properties of the parabola to calculate the area.
Consider the chord connecting points (0,0) to (1,1). The equation of the chord is yx. The points of intersection of the parabola xy^2 with the chord yx can be found by solving:
[ x y^2 ]
[ x x^2 ]
Which gives us the points (0,0) and (1,1).
To find the vertices of the area, we need to find the intersection of the tangent lines. By using the discriminant method, we find the tangent line equation, and then use it to find the vertices.
The area can also be found using the Shoelace formula for a more geometric approach. However, the integral method provides a straightforward and accurate result.
Summary
The area between the parabolas yx^2 and xy^2 in the interval [0,1] is (frac{1}{3}) units2. This can be calculated using integration or symmetry. The integral method provides a precise and efficient approach, while geometric methods offer alternative perspectives.
Keywords
area between parabolas, integration, symmetry, square roots, calculus