Parabola - Wikipedia
2 days ago · The area A of the parabolic segment enclosed by the parabola and the chord is therefore =. This formula can be compared with the area of a triangle: 1 / 2 bh. In general, the enclosed area can be calculated as follows. First, locate the point on the parabola where its slope equals that of the chord.
11.5: Conic Sections - Mathematics LibreTexts
Jan 17, 2025 · The general form of a parabola is written as \[ax^2+bx+cy+d=0 \label{para1} \] or \[ay^2+bx+cy+d=0.\label{para2} \] Equation \ref{para1} represents a parabola that opens either up or down. Equation \ref{para2} represents a parabola that opens either to the left or to the right.
15.2: Double Integrals over General Regions - Mathematics …
Jan 17, 2025 · We can use this idea to define a double integral over a general bounded region. To evaluate an iterated integral of a function over a general nonrectangular region, we sketch the region and express it as a Type I or as a Type II region or as a union of several Type I or Type II regions that overlap only on their boundaries.
6.1: Areas between Curves - Mathematics LibreTexts
Jan 17, 2025 · We start by finding the area between two curves that are functions of \(\displaystyle x\), beginning with the simple case in which one function value is always greater than the other. We then look at cases when the graphs of the functions cross. Last, we consider how to calculate the area between two curves that are functions of \(\displaystyle ...
Conic Sections | Brilliant Math & Science Wiki
5 days ago · A mirror that has a cross-sectional parabolic shape has the property that all light shone directly towards it will be reflected towards the focus of the parabola. A parabolic reflector is a mirror or reflective device that uses this property …
Conic Sections | Circle, Parabola, Ellipse, Hyperbola - GeeksforGeeks
Jan 15, 2025 · Parabola; Hyperbola; Ellipse; Each type of conic section has unique properties and equations, making them essential for understanding orbital mechanics, designing optical systems, and solving quadratic equations
Elliptic curve - Wikipedia
Jan 9, 2025 · Elliptic curves are especially important in number theory, and constitute a major area of current research; for example, they were used in Andrew Wiles's proof of Fermat's Last Theorem. They also find applications in elliptic curve cryptography (ECC) and …
Ellipse - Wikipedia
2 days ago · A general ellipse in the plane can be uniquely described as a bivariate quadratic equation of Cartesian coordinates, or using center, semi-major and semi-minor axes, and angle
Area of a Surface of Revolution
Jan 1, 2025 · The equation of a semi-circle in Quadrants I and II is given by f (x) = √ r 2 − x 2. Because this function is symmetric about the y-axis, consider rotating only the Quadrant I portion of the semi-circle about the x-axis to generate half a …
6.4: Arc Length of a Curve and Surface Area
Jan 17, 2025 · If a rocket is launched along a parabolic path, we might want to know how far the rocket travels. Or, if a curve on a map represents a road, we might want to know how far we have to drive to reach our destination.