Consider the family of surfaces Sa (where is a parameter such that 0 < a < n/2) with equations 2² + y? – sin°(@)z² = cos²(a). (a) For each with 0 < a < 7/2, Sa is a quadric surface. Name the surfaces appearing in the family Sa along with the corresponding range of a values for which they arise. (b) Show that the lines with parametrizations r(t) = (cos(a), sin(2a)t, ±2 cos(a)t) lie com- pletely in Sa. (c) The line r†(t) arises as (a component of) the intersection of a plane with Sq. Find the equation of this plane (in terms of a) under the condition that it intersects Sa in a right angle and a £ T/2.

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Q: (iii) dz< 22 + 2 where here y parametrises the straight line segment from 0 to 1- i.

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A: Perametric equation .

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.