Let G be the solid enclosed by the plane z= y, the xy-plane, and the parabolic cylinder y = 1-x². If ch(x) dV = 0. then g(a,2b)+h(2b)= 6- -13 - 14 -9- -5 -2 -7

Q: Find a plane containing the point (7,3,5) and the line x(t) - 3 + t | y(t) 8 +t z(t) - 3 + 5t

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Q: If 7(t) = is not smooth on R, then b а. 2 b. 4 C. -3 d. -2 е. 3

A: Disclaimer: Since you have asked multiple questions, we will solve the first question for you. If…

Q: x(t) Find the point at which the line y(t) z(t) = = = 4 - 8t -4+ 5t intersects the yz plane. 6 + 2t

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Q: Determine the intersection of 3x – 4y – 5z = 0 with [x, y, z] = [3, 1, 1] + t [2, -1, 2].

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Q: 2. (§1.1) Sketch the following set of points in the x-y plane. {(x, y) = R²: (y - x²) (y + x²) < 0}.

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Q: Advanced Math Question

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Q: (c) If f is a C² scalar function, then V × (Vƒ) = 0. True False (d) The lines 1₁ (t) = (1 − t, 1+ t,…

A: (C) If c is scalar function Means      curl(grad ƒ)  is = 0   Therefore correct ans is  True

Q: Prove that the line x +1 = 글(-6) - lies in the plane 3x + y – 2 = 3.

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Q: I3D Find the coordinates of the point (x, y, z) on the plane z = 2 x + 2y + 2 whi X = y =

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Q: sin(7y), 8x2+7x°cos(7y)). vative servative n of the parabola y=x2-16 from (4,0) to (-4,0). Which of…

A: We have to choose the option

Q: x (b) Hyperbolic paraboloid. M: z = a Show that M is covered by the single patch х(и, v) — (а(u+ v),…

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Q: Show that ƒ(z) = 3x² + 2x − 3y² − 1 +i (6xy + 2y) is differentiable at every point in the plan -

A: The given function is f(z)=3x2+2x-3y2-1+i(6xy+2y)...(i).

Q: The points on the surface x2−2y2−3z2 = 33 at which the tangent plane is parallel to the 8x+4y+6z=5…

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Q: Find the point at which the line (-3, 3, 0) + t(-2, –3, –2) intersects the plane -2x – y + 3z = 5.

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Q: Find a scalar equation of the plane containing the line x = 4+ 5t, y = 5+3t and z = 6+ 4t the point…

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Q: The line L(t) = (3+t, 12+5t, -12-5t) intersects the plane 5x - 5y = 15 at what point?

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Q: Show that the line x + 1 =y/−1=(z − 2)/2 is in the plane 2x + 4y + z = 0.

A: Given equation of a line, x+1=y-1=z-22 And equation of a plane, 2x+4y+z=0

Q: The line L(t) = (4 - 3t, 13 – 3t, 4t – 10) intersects the plane 2z - (4x + 2y) = 16 at the point |…

A: We have to given  line L(t)=&lt;4−3t,13−3t,4t−10&gt; which intersects the plane 2z−(4x+2y)=16 at the…

Q: (a) Find two points on the line x = -1+3t, y=t + 2, z= 2t – 7. (b) ( Find The point at which the…

A: Equation are given in parametric form: Vary parametric value for different points.

Q: The line L(t) = (4 – 3t, 3, 1 +t) intersects the plane 3x – 3y + 2z = -16 at the point when t =

A: To find the point of intersection of the line and the plane, substitute the values of x,y, z from…

Q: Find the point at which the line (4, –1, 4) + t(4, 2, 3) intersects the plane 4x 5y + z = 88.

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Q: Among all the points on the graph of z= 12-x² - y² that lie above the plane x + 4y +62 = 0, find the…

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Q: Find /| F-dS if F(x, y, z) = 2xi+2yj+2zk and S is the part of the paraboloid z = = 8-x?-y? lying…

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Q: Determine which of the following points are in the plane 3(x − 2) + 2(y + 3) − 4(z − 4) = 0: (a) (0,…

A: To check whether the given points are in the plane or not, substitute given values in the equation…

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: