If the curve y = f(x), a ≤ x ≤ b, is rotated about the horizontal line y = c, where f(x) ≤ c, find a formula for the area of the resulting surface.

Q: Find the equation of the tangent plane to the given surface at the specified point. z=xsin(x+y),…

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Q: If (a,ß.r) is a point at which the surface x2+y2-z-2x+42=0 has a horizontal tangent plane, then Iyl=

A: Horizontal tangent plane

Q: The horizontal tangent plane of the surface z = 2x² – xy – y² + 6x – 6y – 2 is given by the equation…

A: Consider the given equation. z=2x2-xy-y2+6x-6y-2 Find the derivative with respect to x and y.…

Q: The area of the surface obtained by rotating the curve y = 22 x + 1, 0 < x < 44 about the x-axis is

A: Consider the given curve. y=22x+1 in the interval 0≤x≤44 The problem requires to find the area of…

Q: Find an equation of the tangent plane to the surface g(x, y) = x2 + y2, at the given point (1, −1,…

A: We have given the surface; g(x, y) = x2 + y2 and the point (1, −1, 2)

Q: Find the point(s) on the surface z = 3 − x2 − y2 + 6y, at which the tangent plane is horizontal.

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Q: Find the exact area of the surface obtained by rotating the curve about the x-axis y = v5 – x, 3 <x…

A: Given:- y=√(5-x) , [3, 5] To find:- Area of surface rotating about x axis

Q: Find the slope of the tangent line to the curve that is the intersection of the surface z = x2 - y2…

A: Calculate point of intersection and differentiate it to get the answer

Q: Find the point on the surface z=x2-y2 at which the tangent plane is parallel to the plane…

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Q: Find the exact area of the surface obtained by rotating the curve about the x-axis. x3 y = 2 1 6x' 2

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Q: The area of the surface obtained by rotating the curve y = x2, 0 <x< 12 about the y-axis is

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Q: Given the surface z = In(-2x² + 5y² – xy). Find the directional derivative of the function f at…

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Q: The horizontal tangent plane of the surface z = 4x – xy – 2y – 2x – 8y – 3 is given by the equation…

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Q: 2. For the surface z = In(y+ Vx² + y²). (a) Find an equation of the tangent plane at the point (3,…

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Q: Consider the surface z = 2 – xy in R³. Use the second derivative test to find the points on this…

A: Given curve z = 2-xy To find :- Use the second derivative test to find the points on this surface…

Q: Suppose that an equation for the surface is S(x,y) = xye^x. In what direction (relative to the…

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Q: Q3. A- Use Greens theorem to evaluate the cireulation or flus for the velocity fiekd : S(x, y)=- *+y…

A: Given velocity field is  f(x,y)=1x+yi-1x+yj we have to find the flux for the boundary of square…

Q: At which of the points below is the tangent plane to the surface x² – 2y + z + yz = 2 vertical?…

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

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Q: At which of the points below is the tangent plane to the surface z - 2y + z?+ yz = 2 vertical?…

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Q: Find the angle of inclination of the tangent plane to the surface x2+ y2+ z2 = 14 at the given point…

A: Consider the function fx,y,z=x2+y2+z2-14. Gradient of f can be obtained as follows. ∇fx,y,z=∂∂x,…

Q: Find the area of the surface generated when the given curve is revolved about the x-axis. y = V3x +5…

A: Surface area of the bounded region along x axis

Q: Find the plane tangent to the given surfaces at the specific point f(x, y) = x2 ln(2x+y) in (-1, 3)

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Q: Find the point on the surface f(x,y)=x²+y²+xy-17x+9y at which the tangent plane is horizontal.

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Q: 1.a)Find the area of the surface generated by revolving the curve y=√(121-x^2), −1 ≤ x ≤ 1 about the…

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Q: Find an equation of the tangent plane to the given surface at specified point. a) z = e*y +…

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Q: Find the point on the surface f(x,y)=x2+y2+xy-26x+29y at which the tangent plane is horizontal.

A: For Given surface f(x,y) The slope of horizontal tangent plane is equal to zero.

Q: Find the area of the surface generated when the given curve is revolved about the y-axis. y = (x^2…

A: Solution:

Q: The area of an ellipse with axes of length 2a and 2b is nab. The percent change in the area when a…

A: Area of ellipse initially  = Ai  = πab    Now, lengths of axis are changed,   They are  [ 2(1.0099)a…

Q: Find the sum of the x-intercept, y-intercept, and z-intercept of any tangent plane to the surface…

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Q: 35. Find the area of the surface generated by revolving the para- metric curve x =t, y = 21² (0 <…

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Q: Find an equation of the tangent plane to the given surface at the specified point z = √√√xy, (3, 3,…

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Q: Find a point on the surface z = 3x^2 − y^2 at which the tangent plane to this surface is parallel to…

A: We need to find the Point on the surface of  z=3x2−y2 given tangent plane to this surface is…

Q: Find the point on the surface f(x,y)=x+y2+xy-24x+2y at which the tangent plane is horizontal.

A: We have to find the points for horizontal tangent plane.     According to our guidelines , We can…

Q: Find an equation of the tangent plane to the given surface at the specified point. f(x,y) =…

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Q: (d) Find the area of the surface generated by revolving the curve y = 2a, 0 < x < 3, about the…

A: Part (D) To find the area of the surface generated by revolving the curve, y=2x,0≤x≤3, about the…

Q: Find the area of the surface generated by revolving the given curve about the x-axis. y = V9 -x,…

A: 3

Q: The area of the surface obtained by rotating the curve y = √√x + 1, for 0 ≤ x ≤ 2, about the x-axis,…

A: Given :

Q: The area of the surface obtained by rotating the curve y = V4-x², -2 s xs 2 about the x-axis is

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Q: Find an equation of the tangent plane to the surface at the given point. h(x, y) = In V x2 + y², (6,…

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Q: If (a,B,y) is a point at which the surface x²+y²-z²–2x+104=0 has a horizontal tangent plane, then…

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Q: If 0 ≤ f (x) ≤ g(x) for x in the interval [a, b], can the surface obtained by rotating the graph of…

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Q: (b) Find the point that makes the tangent plane to the surface z = x- xy +y-2x + 4y horizontal.

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Q: 35. Find the area of the surface generated by revolving the para- metric curve x = t, y = 21² (0 < 1…

A: Given: x=t y=2t20≤t≤1

Q: メ y= oL Xとn2

A: Given curve is y=ex , 0≤x≤ln2find the area of the surface

Q: 35. Find the area of the surface generated by revolving the para- metric curve x =1, y= 21² (0 <…

A: Given: The curves,  x =t,  y = 2t2,   0 ≤ t ≤ 1 To find: The area of the surface generated by…

Q: Find an equation of the tangent plane to the given surface at the specified point. z = y In(x), (1,…

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Q: Find the exact area of the surface obtained by rotating the curve about the x-axis. y = V8 - x, 2 s…

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Q: The exact area of the surface obtained by rotating the following curve 6x = y + 12, 2 < x < 3 about…

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Q: The tangent plane to the graph of the surface z=In(x2+y²) at the point (1,0,0) contains point…

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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.