Let Dxy be the region in the xy-plane enclosed by the lines y=2−x,y=4−x,y=x and y=0. Let (x,y)=F(u,v)=(u+uv,u−uv). Use the mapping F to evaluate the following integral. ∫∫Dx e^x−y/x+y/x+y dxdy=   Hint: If F−1 exists on Dxy you should be able to determine the image Duv. If all is well then the image of the integrand will be much easier to compute and the result should follow from the change of variable theorem.

Q: Evaluate F. dr using the Fundamental Theorem of Line Integrals. Use a computer algebra system to…

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Q: Evaluate the line integral x¹ z ds, where C is the line segment from (0, 7, 5) to (7, 1, 2). C

A: We have to evaluate  where  is the line segment from  to

Q: Calculate the line integral [F•dr, where F = 3x°yi + x'j in two cases for C1 and C2. These two…

A: The given vector is F=3x2yi+x3j Here we use Green's theorem. So the Green's theorem is…

Q: evaluate the integral

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Q: Let f(x, y, z) = x^2 + y^3 + z^5 and let u =. Find the directional derivative Du f(x, y, z).

A: The directional derivative Du​f(x,y,z) is 64/81​.Explanation:Step 1: To find the directional…

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.