use the inner product < p,q >= p(x)q(x)dx. to find the projection of p(x) q(x) = 5x + 2. = S'; x3 onto

Q: Compute the double integral of (x – 2y) Væ + 2y o i over the quadrilateral with vertices (0, 0), (4,…

A:

Q: Find the line integral of x ds over C where c is the line segment from (1,0) to (0,1) using two…

A:

Q: Evaluate the line integral / (x+ 2- rom (0,0) to (3, 0), the path C2 fro co (0,0) by applying the…

A: Consider the provided question, Hello. Since your question has multiple sub-parts, we will solve…

Q: 4. Calculate the integral (x – 3y) dA where R is the triangular region with vertices (0,0), (2, 1)…

A:

Q: Calculate the line integral f -y²dx + xdy where ŕ is the path y² = 2x − x² such that x>1 and y > 0.

A:

Q: Show that the integral is independent of the path, and use the Fundamental Theorem of Line Integrals…

A: Given:

Q: 3. Show that the line integral 4. x + y)dx + dy C is independent of path and evaluate this integral…

A:

Q: Use the specified substitution to find or evaluate the integral.

A: We need to use the specified substitution to find or evaluate the integral.

Q: Verify Green's Theorem by evaluating both integrals √ y² dx + x² dy = √ √ ( Ox (ON-OM) c for the…

A: Here we have to verify the Green's theorem.

Q: Evaluate the line integral §. xy²dx + 2x²y dy over triangular path with vertices (0,0). (2,2) and…

A:

Q: Evaluate the line integral (x+2y)dx + dy, where C consists of the path C from (0,0) to (3,0), the…

A:

Q: Find the Jacobian for the transformation x = 3u + v and y = u+ 3v. NEXT, use this transformation to…

A:

Q: and y = 2u, evaluate the integral xy³dA where c) Using the transformation x =-; 6u region R is…

A:

Q: Use Green's Theorem to evaluate the line integral |(y - x)dx + (2x – y)dy for the given path. C: x =…

A: Let's find.

Q: Show that (2x In yz-5ye") dx-(5e* -x²y)dy +(x²z- +2=) dz, where C runs from (2,1,1) to (3,1,e), is…

A:

Q: Evaluate the integral ∮C 4x3y dx + x4 dy for any closed path C.

A:

Q: Evaluate the line integral f. (2x – 4xy)dx + 3xy – 2x³y)dy over the straight path L joining point…

A:

Q: The solid bounded by the paraboloids 2= - 8+z²+ y² and z -3-32²-3y² has volume:

A:

Q: Evaluate the line integral §. xy²dx + 2x²y dy over triangular path with vertices (0,0), (2,2) and…

A:

Q: Evaluate the given line integral. I (y, –x) · dR, where C consists of 4y = x² from (-2, 1) to %3D C…

A: Line integration of the function

Q: Carry out the affine transformation | u = (x + y)/V2 lv= (y - x)/V2 in the improper double integral…

A:

Q: (z+ y – 1) cos D

A:

Q: Evaluate the triple integral xy dV where E is the solid tetrahedon with vertices (0, 0,0), (6, 0,…

A:

Q: Use the transformation u = y - x, v = y, to evaluate the integral on the parallelogram R of vertices…

A:

Q: Find the product of the integral 1- S x²cos(xy)dxdy

A: We have to change order of integration first.

Q: Evaluate ∫∫∫E √x2 + z2dV, where E is the region bounded by the paraboloid y = x2 + z2 and the plane…

A: Given, y=x2+z2 ⇒z2=y-x2z=±y-x2 Therefore the limit of z is -y-x2≤z≤y-x2 On drawing the graph in the…

Q: Evaluate the iterated integral: dydx - 1/4 Jx2

A:

Q: Use Green's Theorem to evaluate the line integral √(x - (y - x)dx + (2x - y)dy for the given path.…

A: Let's find.

Q: Use the transformation u = - -6x +y, to evaluate the integral [ ( (5x • y) dx dy, where R is the…

A: use the transform u=-6x+y, v=5x+ywe evaluate ∫∫R 5x+y dx dywhere R is the parallelogram bounded by…

Q: Find the line integral of F =9xy*ï +(x² + 3),yj over the path L.

A:

Q: Let xbe inner Product and ||x1|=|ly|| = 1 x+y Lix.y ? why?

A: Given that X is an real inner product space. x ,y be two unit vector from X then need to show that…

Q: Let T be the triangular region bounded by the lines y = x,0 <x <1. By using the transformation x = u…

A: Let T be the triangular region bounded by the lines y=x, 0≤x≤1. Given transformation x=u+v and y=u-v…

Q: Use Greens Theorem to evaluate: §. (2x + ev´)dy - (4y2 + e*")dx where C is the boundary of the…

A:

Q: Use the transformation u = y - x, v = y, to evaluate the integrals on the parallelogram R which has…

A: We need to evaluate the integral: ∬Ry-xdA using the transformations: u=y-x, v=y. Here, R is a…

Q: Use Green's Theorem to evaluate the line integral (y – x)dx + (2x – y)dy for the given path. C: x =…

A: Given integral is ∫Cy−xdx+2x−ydy where C:x=2cosθ,y=sinθ,0≤θ≤2π. Green theorem is given by…

Q: Let T be the triangular region with vertices (-1,0), (0,1), and (1,0). Use an iterated integral to…

A:

Q: Verify Green's Theorem by evaluating both integrals ам | y2 dx + x2 dy dA ду for the given path. C:…

A: Solution is given below:

Q: Evaluate the line integral di the path over C from to Q(2,4) P(O,) where F(xy) 2xy°2+3y5 13D to ネソーメ…

A:

Q: 15. Evaluate the line integral J.(3xy* - 6xy ]dx + (4x°y³ - 3x?) dy, where C is any path from (1,2)…

A:

Q: Compute the path integral of F = ⟨ y , x ⟩ along the line segment starting at    ( 1 , 0 ) and…

A:

Q: Evaluate the line integral y² dx + x²y dy where C' is the rectangle with vertices (0,0), (5,0), (5,…

A:

Q: Compute the line integral of F = (-y, x) along the line segment from (0,0) to (1,1).

A: The line integral is evaluated using the following formula  ∫cF→x,yds=∫abF→rtr'tdt    , a≤t≤b   ,…

Q: is Theorem 1.6. The line integrals of f in D are independent of path if and only if f has an…

A: Given: The line integrals of f in D are independent of the path if and only if f has an…

Q: #1. Evaluate (2x – 5y) dA, R where R is the parallelogram with vertices (0,0), (2,3), (–1,4) and…

A: The given region R is a parallelogram with vertices as shown in below graph. We have to use change…

Q: Evaluate the path integral between the points A = (1,3) and B = (2,5) of: B *(x²y + 2y)dy, with y =…

A:

Q: Evaluate the given line integral. | (y, -x) dR, where C consists of 4y = x² from (-2, 1) to (2, 1)…

A:

Q: Compute the following Cauchy principal value integral: dx р.v. x4 – 1 п/2 -T/2 Зп /2 -27

A:

Q: Evaluate the improper intergal 1 dx using the residue (x² + 1)3 theorem.

A:

Q: (B) Compute div F and curl F for F= x²yi- (z3-3x)j + 4y2k %3D

A:

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.