9. The Whitney umbrella surface is given parametrically by I= uv, y = v, 2= u. A picture of this surface is:

Q: What is the equation of the plane to the surface z = e3y sin(3x) +2 at the point P ,0, 3 %3D O x- y…

A: The solution is given in the next step.

Q: (7) Let D be the solid above the cone z = r and below the sphere of radius 2, and let fla, 4, 7) =…

A: To convert the cartesian coordinates into cylindrical coordinates put  And the integral will be…

Q: Given S as a quadric with equation 4x^2 - (y-3)^2 + 4z^2 = 4 Find a generating curve for S (as a…

A: Given S as a quadric with equation 4x^2 - (y-3)^2 + 4z^2 = 4 Find a generating curve for S (as a…

Q: A torus of radius 2 (and cross-sectional radius 1) can be represented parametrically by the function…

A: The surface area (A) of torus is given by the formula mentioned below with given values as.

Q: Which of the following sets of parametric equations is for the surface obtained by rotating th curve…

A:

Q: 2) The (Beltrami) pseudosphere is given by the chart x Cu,v) =( Coscul Sin(v), Sincul Sin(vl,…

A:

Q: 31. Use sine and cosine to parametrize the intersection of the surfaces x² + y² = 1 and z = 4x²…

A: Given x2+y2=1 & z=4x2 x2+y2=1 is the equation of circle. That is x2+y2=12,  Therefore, Radius is…

Q: Question 9 Which surface has the following equation? a2-2y = 32 O Cylinder O Plane O Paraboloid O…

A:

Q: (b) A simple way would be as follows. Let z=t, where 2<t< 9. It follows that * + y = t-1. We can…

A: Given function is 1 + x2 + y2 - z = 0 for 2≤ z≤9 Now we have to find the parametric representation…

Q: sketch the solid whose spherical coordi- nates (p, 4, 0) satisfy the given inequalities. 0<p<2/cos…

A:

Q: 15. Find the radius of curvature at 0 = ÷n of the parametric equations x = 3(0 + cos 0), y = 3(1 —…

A: Given two parametric equations, x=3(θ+cosθ),y=3(1-cosθ) To find the radius of curvature at θ=π3

Q: 4. Graph the surface 1 = y² + z² – by first sketching at least three cross sections in each of the…

A: Graph the surface by using a 3D graphing calculator like desmos. Three cross sections in xy plane…

Q: Use Green's theorem to evaluate $(xy + 3x²)dx + (3y – 2x)dy, along C, where Cis the triangle with…

A: Given Integral ∮(xy+3x2)dx +(3y-2x)dy along c where c is the triangle with vertices (0, 0), (0, 3)…

Q: a. Which of the following surface graphs represents f(x,y) = y²? OA.

A: Given, the surface                           z=f(x, y)=y2

Q: The attached figure shows the surface S1: y + z= 2, limited in the first octant with surface S2: x =…

A: The area of a surface z=f(x,y) bounded by the region R is given by the integral expression:-…

Q: In three dimensions, the cylinder x? + z2 – 10x – 6z – 2 = 0 has radius k i +tj+ and the axis line…

A: Radius =6  r(t)= (5+6cos(thetha))i+tj+(3+6sin(thetha)k

Q: Find the equation of the sphere of radius 8 centered at (1, 3, 6). O x² + y² +z² = 46 O x² + y² + z²…

A:

Q: If (a,B.y) is a point at which the surface x2+y2-z²-2x+12=0 has a horizontal tangent plane, then y|…

A: As per guidelines, we will solve first question only. Let α,β,γ be the point on the surface…

Q: If (a,B,y) is a point at which the surface x-+y²-z²-2x+12=0_has a horizontal tangent plane, then…

A:

Q: 1. Find the area of the part of the surface z = a? + 2y that lies above the triangle with vertices…

A:

Q: What is the equation of the plane to the surface z = e3y sin(3x) +2 at the point P ,0, 3 O x - y -…

A:

Q: 3. Consider Hhe following two surfaces. 8x-54-82 = -13 a) Show that the surdaces intersect at the…

A: Given surfaces are  z=2 xy2 and 8x2-5y2-8z=-13 (a)  Let fx,y,z=z-2 xy2=0 Let gx,y,z=8x2-5y2-8z+13=0…

Q: 4. Consider the surface defined by the portion of the cone z = -Vx2 + y? that lies below the…

A:

Q: The equation of the tangent plane to the surface yey +z =0 at the point (0,-1,1) is az+by+cz = 1.…

A:

Q: 6. Consider the sphere a2 + y? + (z-3)2 = 1. Find the point P (a, b, c) on the sphere such that the…

A:

Q: 2. For the surface z = In(y+ Vx² + y²). (b) Find the linear approximation at the point (3,-4).

A:

Q: .2 The surface z? = 1 is a 25 - - 36 hyperboloid of one sheet. Select one: OTrue OFalse

A:

Q: Consider the parametric surface S given by the portion of the sphere x +y² +z² = 4 where z 2 1.

A: In this question, we have to parameterize the surface in cartesian coordinates and then find the…

Q: Which type of surface does the curvex= cos(3t),y= sin(3t),z=t, lie on?

A:

Q: ded above by surface z = 10 – x2, bou e planes y = 0 and y = 3.

A:

Q: 28. Show that the ellipsoid 2x² + 3y² + z² = 9 and the sphere %3D x? + y? + z? - 6x – 8y – 8z + 24 =…

A:

Q: 3.4.8 Show that the sphere x² + y² + z² = 1, z ≥ 0. JJ,z²dS=, where S is the surface of

A:

Q: What is the equation of the plane to the surface z = e3Y sin(3x) +2 at the point P,0, 3) O x- y + ez…

A:

Q: If a parametric surface given by r1(u, v) = f(u, v)i + g(u, v)j+ h(u,v)k and –1< u<1, –2 < v< 2, has…

A: We will find out the required value.

Q: 14. Under the paraboloid z=x² +y2 and above the disk x +y² <9 for (x,y) in the first quadrant.

A: we have to find the volume

Q: b) a dr+(x+yz) dy+(ry- VE) dz, C is the boundary of the part of the plane 3r+2y+z in the first…

A:

Q: 1-) Given the surface z = In(-2x2 + 5y2 – xy). Find the equation of the tangent plane and the…

A:

Q: 2) The (Beltrami) pseudosphere is given by the chart x(uivl = (coslu) SinCv), Sincul Sincv), coscv)+…

A:

Q: If a parametric surface given by ri (u, v) = f(u, v)i + g(u, v)j + h(u, v)k and -3 <u < 3, –4 < v<…

A: We solve this

Q: 1. Identify the parametric surface S given by: r(u, v) = 3 cos ui+ vj + 3 sinu k where 0 <u < 2n and…

A:

Q: Given the surface z = In(-2x² + 5y² – xy). Find the equation of the tangent plane and the parametric…

A: Surface given z=ln(-2x2+5y2-xy) Point P4,3,0

Q: 7. Consider the Surface S:z = 52² + 5y?. An equation of the plane tangent to Sat the point P (3, 3,…

A:

Q: If o is the sphere of radius 2 centered at the origin, then (x² + y² + z³) d$ =

A: Given the sphere

Q: Consider the spherical surface E with center at C= ( 4, 3, 2 ) and radius 4 . Let π be the tangent…

A:

Q: 3. Show that every sphere through the circle x2 +y-2ax+r²=0, z=0 cuts orthogonally every sphere…

A: This is a co-ordinate geometry problem

Q: SSVitzý +Z% dA and then converting to polar. This The surface atea of a sphere ot radius I is As=4r.…

A:

Q: Consider the surface described by x² – v² = 144. Which of the following describe the same surface in…

A:

Q: Are there any points on the hyperboloid r² – y² – 2² =1 where the tangent plane is parallel to the…

A:

Q: 15) Consider curve C which is the intersection of the surfaces shown in the attached figure. The…

A: The equation of the graphs given are as follows: x=-34y2+3y=yz=y22 We are asked to evaluate the…

Q: (3) Consider the quadric surface S C R, having equation a2 + y? - z2 = 9. Find all lines through P…

A: General equation of line from point 3,-3,3 is x,y,z=3,-3,3+tl,m,n.  Substitute it in the quadratic…

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.