Let f(x) = 2x - 1 and g(x) = x - 6 on [ 6, 10 ]. Find the center of gravity of the region between the graphs of f and g. center of gravity = (x, y) where

Q: Determine the equation of the normal line to the surface In (x + y) = Z 2 at the point (-1,2,0)

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Q: Evaluate | f(x,y) ds where Cis the portion of the curve y = In x from the point (1,0) to the point…

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Q: Let f(x.y) = ry. a. Sketch/describe the domain of g Domain = { (x,y)]yz0} b. Sketch a contour map…

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Q: X- An ant is moving on the surface z = x² + xy + y² such that its - and y-coordinates satisfy cost,…

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Q: - Find div F and curl F if F(x, y, z) = 10y¹z7i – 17xªzºj – 3xyễk. div F= curl F=

A: The general form of a vector-valued function in three dimensions is given by F=F1i+F2j+F3k, where…

Q: n exact form. Use symbolic notation and fractions where needed.)

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Q: Find the absolute maximum and absolute minimum values of the function f(x, y) 2x3 + 3xy + 2y %3D…

A: We have to find the absolute maximum and absolute minimum values of the function fx,y=2x3+3xy+2y3…

Q: Find the symmetric equations for the normal line LN to the surface S:f (x, y) = 5Vx2 + y? at the…

A: if  f is  function of (x,y,z) then gradient of f is defined asfx i + fy j + fz k ∇f is the normal to…

Q: Plot the slope field of y=x+y using slopes m = -3, -2,-1,0, 1, 2. Sketch integral curves for…

A: First we plot the slope field and then plot the curve .

Q: - Integrate the function f orgy) =y-Vx over triangle cut from the first quadrant by Line X+y=1

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Q: Find the centroid of the thin plate bounded by the graphs of g(x)=x² and f(x) = -x + 56. Use the…

A: The region is bounded by the graphs fx=x2 and gx=-x+56. We have to find the centroid of the region.

Q: Suppose F(x,y)=. Use Green's Theorem to calculate the circulation of F around the perimeter of the…

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Q: Find extrema of f (x, y) = 2x² + 4y² on the unit-circle x + y² = 1.

A: To find the extrema of the function ƒ(x, y) = 2x² + 4y² on the unit-circle x² + y² = 1, we need to…

Q: Integrate the function f(r,y, z) = Over the entire 3D coordinate system, i.e., all of R'.

A: We have to integrate the function fx, y, z=1x2+y2+z21+x2+y2+z2 over the entire 3D coordinate system,…

Q: Evaluate e [ f(x, y, z) ds, where f(x, y, z) = 2 + 2y + (z + x) and C is the line segment from (2,…

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Q: of Let f(x,y) = xy° +7. Evaluate at (x,y) = (-7,1) and interpret the results. ду -

A: In this question we have find δfδy at (x, y)=(-7, 1)

Q: Find the absolute extrema of the function f defined by f(x) triangular region R with vertices (0,…

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Q: Find the centroid of the thin plate bounded by the graphs of g(x)=x² and f(x) = x + 6. Use the…

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Q: Let X ~ Uniform(-1,1) and Y = X² , find the CDF and PDF.

A: Use the following definitions, to obtain the solution. The cumulative distribution function of a…

Q: Let F(x, y, z) y³ 2:5 . What is the value of div F at (1, 2, 3)?

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Q: Find the points on the graph of the function f(x) = x^2 that are closest to (0, 2) in the xy-plane.…

A: Minimal distance

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.