iminate the parameter in x = -t +2t and y =t-4 and then identify the parametric sketch its image in the xy-plane on a piece of paper.

Q: Find an equation of the tangent plane to the given parametric surface at the specified point. x = u…

A:

Q: Eliminate the parameter t from the parametric equations x = 3 cos t, y = 5 sin t and then sketch the…

A: In eliminating parameter t from the given parametric equations we do establish the relationship…

Q: Find a set of parametric equations for the tangent line to the curve of intersection of the surfaces…

A:

Q: Sketch the curve defined by the parametric equations x = t2, y = t + 1, -∞ < t < ∞.

A:

Q: 12- Find an equation of the tangent plane to the given parametric surface at the specified point.…

A:

Q: Find the angle between the two planes 6x − 3y + z = 5,  −x + y + 5z = 5 and find a set of parametric…

A: We have given the equation of planes; 6x − 3y + z = 5 −x + y + 5z = 5 Hence the normal vectors of…

Q: Determine A so that the curve y = 4x+ 16 can be written in parametric form as x(t) = t – 5, y(t) At…

A: Put value in given equation and simplify

Q: 4. Find the parametric equations for the tangent line to the curve of intersection of the y? z1…

A:

Q: Write an equation in general form of the tangent plane to the parametric surface at the stated…

A: Equation of tangent plane

Q: use the given graphs of x = ƒ(t) and y = g(t) to sketch the corresponding parametric curve in the…

A:

Q: Find a set of parametric equations for the tangent line to the curve of intersection of the surfaces…

A: Given surfaces are And the given point P is

Q: Sketch the curve with parametric equations x = sin t y = sin2t

A:

Q: Determine the x - y coordinates of the points where the parametric equation x = 3 sint y = 3 cost 0≤…

A:

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

A:

Q: Sketch the parametric curve of the given parametric equations and indicate the orientation. 1. x=3t…

A:

Q: Sketch the plane curve represented by the parametric equations x = Vi and y = 2t – 1 by eliminating…

A: Sketch the plane curve represented by the parametric equations             x=t     and     y=2t-1 by…

Q: Find the exact length of the parametric curve x =1+3t², y = 4+2r°, 0<t<2.

A:

Q: 7. Sketch the curve with parametric equations z =e, y =e', teR.

A:

Q: Determine the x - y coordinates of the points where the parametric equation x = 3 sint y = 3 cost…

A:

Q: Find a parametric equation for the curve of intersection of the cone x²+y² = z² and the plane x+2z =…

A: We have to find a parametric equation for the curve of intersection of the cone x2+y2=z2 and the…

Q: Sketch the curve defined by the parametric equations x = sin πt/2, y = t, 0≤ t ≤ 6.

A:

Q: Determine the parametric equations of the curve of intersection of the paraboloid x? – y? + z² = 0…

A: Detailed explanation mentioned below

Q: iminate the parameter in the parametric equations x(f) = 5t – 8 and y(t) = Vt – 5 to find a…

A: Given: The parametric equations are xt=5t-8 and yt=t-5 Given query is to find the Cartesian…

Q: Without eliminating the parameter t, find a Cartesian equation for the lin curve C, defined by the…

A:

Q: Find an equation of the tangent plane to the given parametric surface at the specified point. x = u…

A:

Q: Complete the parametric equations for the line where the planes 8x + 11y + 4z 13x + 2y + 3z = 5…

A: This question can be solved using the concept of lines and planes in three dimensional geometry.

Q: Without eliminating the parameter t, find a Cartesian equation for the line tangent to the curve C,…

A: Let's find.

Q: Find the length of the parametric curve x = e^t + e^-t, y = 5-2t on the interval [0,3]

A: Find the length of the parametric curve x = e^t + e^-t, y = 5-2t on the interval [0,3].

Q: Determine the relative maximum, relative minimum, and point of inflection without eliminating the…

A: The parametric equations are, x=1t3 y=t2-t   Differentiate x=1t3 with respect to t. dxdt=ddx1t3=-3t4…

Q: Find a parametric description for the curve x=y^2-9 from (0, 3) to (-5, -2) such that t=0…

A: We need to parametrize the curve x=y2-9 from 0,3 to -5,2 We need to put either x=t or y=t and see a…

Q: Find the length of the curve having a parametric equation of x= 2cos³0 and y=2sin²0 from 0-0 to…

A:

Q: Sketch the plane curve represented by the parametric equations x = 6 cos t, y = 4 sin t, π ≤ t ≤ 2π…

A: Given  x = 6 cos t, y = 4 sin t, π ≤ t ≤ 2π

Q: a) Find the are length of the parametric curve x = B(t. sint), y = B(1+ cost), in the range 0 sts,…

A:

Q: express the parametric curve in the form y = f (x) c(t) = (4t − 3, 10 − t)

A:

Q: Find the parametric equation for the line where the planes 7x – 6y + z = 10 and 2x + y – 3z = 4…

A: We have, r(t) = (0, -2, -2) + t(17, 23, 19) The detailed solution is as follows below:

Q: Find the length of the parametric curve x = 2t + 1, y = 3 + 2tž, 0 < t <%.

A: Find the Length of the curve

Q: Eliminate the parameter to find a Cartesian equation of the curve with parametric equations x = e^t…

A: Just put the exponential term value from one equation into the other and easily eliminate the…

Q: Determine the length of the parametric curve given x = 6 sint, y = 6 cost, ㅠ 0 ≤ t ≤=.

A:

Q: Find parametric equations for the line tangent to the curve of intersection of the surfaces tan-1(x?…

A:

Q: 7. Sketch the curve with parametric equations x = e-t , y = e', teR.

A: Topic = Functions

Q: Sketch the plane curve represented by the parametric equations x = Vi and y = }t + 1 by eliminating…

A: The parametric equations are given by x=t   …1 and y=12t+1   …2 Take the square of the equation (1)…

Q: Draw the curve given by the parametric equationsx=t3+t2−t,y=t3+2t2−4t.

A:

Q: Eliminate the parameter to find the Cartesian equation for the given parametric curve. = vt +1, y =…

A: Cartesian equations is x^2 - y^2 = 2

Q: Find the area of the surface obtained by rotating the curve of parametric equations: 6 x = 6t – t, y…

A:

Q: Sketch the plane curve represented by the parametric equations x = 5 cos t, y = 2 sin t, 0 ≤ t ≤ π…

A: x=5cost,  y=2sint Now,…

Q: Eliminate the parameter in the parametric equations x(t) = –7t – 2 and y(t) = Vt – 8 to find a…

A: givenx(t)=-7t-2..................(i)y(t)=t-8..................(ii)we can rewrite 2nd equation…

Q: Find the point at which the line with parametric equations x = 3 + 4t, y = 5 – 2t, z = 4 + 7t…

A: According to question,Given equation of line in parametric form,x = 3+4t,y = 5-2t,z= 4 +7t,Equation…

Q: Find the length of the parametric curve defined by x=r°, y=t- for 0st<1 7.

A:

Q: Eliminate the parameter to find a Cartesian equation of the curve with parametric equations x = et –…

A:

Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: