Find the volume of the solid bound between the paraboloid z = x^2 + y^2 and the plane x + z = 0 as follows. (a) Find an equation describing the projection of the intersection of the plane and the paraboloid in the xy-plane. What is the shape of the curve described by the equation?

Q: how was it determined that the equation of the yz plane is x = 0?

A: The objective is to explain yz-plane equals .

Q: Without using a graphing utility, sketch the surface with equation x2 + y2 − z2 = 0.

A: Surface with the equation

Q: Find an equation of the plane that contains the intersection points of the ellipsoid 9x? + 4y² + 9z²…

A: GIVEN : Ellipsoid 9x2 +4y2 + 9z2 = 36

Q: Find the distance between the skew lines with parametric equations x = 1 + t, y = 2 + 6t, z = 2t,…

A: The distance from a point P(x1,y1,z1) to a plane  is

Q: 8.) Find a parametrization for the surface created by rotating the hyperbola z =-around the positive…

A: Consider the given information. Its purpose is parameterization for the surface by rotating the…

Q: Find a parametric form for the line through the point (8, –6, –3) and perpendicular to the plane 2x…

A:

Q: Sketch the surface given by the equation x2+z2/3+y=3 indicating the points where the surface…

A: Sketch the surface given by the equation x2+z2/3+y=3   indicating the points where the   surface…

Q: 6. Find parametric equations for the line of intersection of the planes 2x – y + z = 4 and 3x + y +…

A:

Q: Find the distance between the skew lines with parametric equations x = 3 + t, y = 1 + 6t, z = 2t,…

A: I am attaching image so that you understand each and every step.

Q: Consider the paraboloid given by the equation: z = x^2 + y^2 Find the equation of the tangent…

A: at point (1, 2, 5)at point (1, 2, 5)

Q: y² - (z²/3) = 1 + (x²/12) sketch the quadric surface. emphasize the traces on the coordinate planes…

A: Given a quadric surface y2-z23=1+x212. YZ trace: When x=0, we get y2-z23=1. This is a hyperbola on…

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.