(b) Find the value of the following integrals: z-1 1- dz, y is the triangle with vertices at the points z -51, (z' +z-2) z = 31, z 2.

Q: 2. VI-x² !! 1 dy dx- dz 0 -1V1-x2

A: Change Cartesian coordinate to cylindrical coordinate by using x = r cos(theta) y = r sin(theta) z =…

Q: 6.4.12 Please help me solve these math practice questions.

A: Area of the curve y= cos(4x)  Where -π/8<x<π/8

Q: 8) By using the double integral in polar coordinate, find ,"[S" e-x²+y>dx]dy

A: The double integral in polar coordinates is given by the formula ∬Rf(x,y)dxdy= ∫ab∫g(θ)h(θ)f(rcosθ,…

Q: 1. Sketch the solid of integration of the given integral. Evaluate the integral. r dV: S is bounded…

A:

Q: 3 2(x²+1)7 Write down an integral that gives the ARCLENGTH of y on the interval from x = 0 to x = 2…

A:

Q: 6. Evaluate the line integral (x? + y² )dx – x dy along the curve C: x? + y² =' counterclockwise…

A: Given: ∫c x2+y2 dx- x dy  We have to Evaluate the given line integral along the curve C: x2+y2=1…

Q: [[(2x + y²) da + 2xy dy], where C is the line segment from (-1,0) to (1, 2) Evaluate in two…

A: The line integral ∫CMdx+Ndy involves the integration over a specific path. In this case, the line…

Q: Calculate the integral (x – y) dx + (x + y) dy I = x2 + y2 where L is along the curve y = cos x,…

A:

Q: Evaluate the line integral by the two following methods. xy dx + x?y3 dy Cis counterclockwise around…

A:

Q: 8. By using the double integral in polar coordinate, find SS e-x*+y*)dx]dy

A: As we know that the polar coordinate: x=rcosθ,y=rsinθdxdy=rdrdθ Given: ∫0∞∫0∞e-x2+y2dxdy So, the…

Q: 1- Assume that x, y E R and z = x + iy E C .Calculate the following integrals using the given…

A: Given :-  Assume that x,y ∈ℝ and z = x+iy ∈ℂ. To determine :-  The following contours :- (b)…

Q: Evaluate the line integral by two methods: (a) directly and (b) using Green’s Theorem.3. ∮C xy dx +…

A: According to Greens theorem :

Q: 4: Use Green's Theorem to evaluate the following integral e dx + (5x+ 6) dy Where C is the triangle…

A: Using green theorem to evaluate the following integral.

Q: Calculate the integral =| [e* sin y – b(x + y)] dx + (e* cos y – ax) dy where a, b are constants, L…

A:

Q: Evaluate the double integral x sin(x + y) dA where R3 [0, 풍] x [0, 플] %3D

A:

Q: Calculate the double integral. 9x sin(k + y) dA, R = 0, 풍핑x [o, 플]

A: First we will evaluate the inner integral wrt 'y' and then evaluate wrt 'x'.

Q: Calculate the iterated integral. 1/2 (y + y2 cos(x)) dx dy

A:

Q: Use Green's Theorem to evaluate the line integral () in(x²+y?) oy 2 arctan Y dx + Inx2 C: x = 9 + 6…

A: Green's theorem is a vector identity which is equivalent to the curl theorem in the plane. Over a…

Q: Calculate the iterated integral integral_0^2 integral_0^4 (6 x ^2y - 3 x) dy dx

A: by solving given Integral

Q: Evaluate the following line integrals in the complex plane by direct integration (not using…

A: Complex

Q: 2. Convert the integral In(r² + y? + 1) dx dy into a polar integral and then evaluate it

A: Explained below

Q: Evaluate the double integral I. y sin(a* )dA R where Ris the triangular region with the three…

A:

Q: 3) S. (3x? – 3yz + 2xz)dx + (3y? – 3xz + z²)dy + (3z2 - 3xy + x2 + 2yz)dz Calculate the curvilinear…

A:

Q: Evaluate the double integral y sin(2³)dA where R is the triangular region with the three vertices…

A:

Q: Use Green's Theorem to evaluate the line integral below. dx + 5xy dy C: r = 1 + cos(0), 0 s0 s 2n

A: Consider the provided line integral ∫Cx2-y2dx+5xydy. Let M=x2-y2, and N=5xy. That implies,…

Q: 9-x2 C: Change to polar and evaluate the double integrals S (x3 + xy?)dydx V9-x2

A:

Q: (b) Evaluate the following integrals: z²+cosz (i) [ezsiniz) + + sin22]dz, C: |z| = counterclockwise.…

A: (i) Given integral is∮Cezsinz+z2+coszz10+sin2zz2dzC: z=π2 counterclockwise.The integral is evaluated…

Q: Find the value of 1-1/16 (x² + y²) dy dx + ſ', [~ ² * (x² + y²) dy de by expressing it as one…

A: From, the given integral, draw the region of integration.

Q: Q21 Verify Green's theorem for $. (xy² + x²)dx + (4x – 1)dy where Cis shown in the figure by (a)…

A:

Q: 2. Express the iterated integral as a double integral in polar coordinates. 1 X 1 Lo dy dx 16/2 x² +…

A:

Q: 10. Use a definite integral to find the exact area enclosed by the x-axis and the parametric curve =…

A: The given parametric curve is: x=2t3+1    and     y=2t-t2

Q: 11-y² x 26)A) Convert the integral J Jax?+ y²)dzdxdy to an equivalent integral in -1 0 0

A:

Q: 3. Consider the line segment L = [-1-i, 1 + i]. Calculate the value of the integral 1₁² z² dz.…

A:

Q: 16–x² Choose the iterated integral that is equivalent to y dy dx in polar coordinates.

A: let x=rcosθ and y=rsinθ dydx=rdrdθydydx=r2sinθdrdθ 16-x2=yx2+y2=42=r2 r=4 Therefore r varies from 0…

Q: Calculate the iterated integral.   Integral_-8^8integral_0^pi/2 (y + y2 cos(x)) dx dy

A:

Q: 6. By using the double integral in polar coordinate, find ff e-(x²+y')dA for the region enclosed by…

A:

Q: Evaluate the line integral along the curve C. (3x² + y²) dx + 2xydy, C: x = cos f, y = sin 1, 0≤1≤ 2…

A: Given integral is ∫C3x2+y2dx+2xydy , where curve C:x=cos(t), y=sin(t), 0≤t≤π2 We need to find the…

Q: Depending on the real parameter p and q, what can you say about the result of the following im-…

A: a ∫1+∞1xpdx=limt→+∞ ∫1t1xpdx     =limt→+∞ x1-p1-p1t   =11-p limt→+∞ t1-p-1   =11-p∞  =∞

Q: Find the line integral of C in two ways; A) Directly. B) Using Green's theorem. C eata formed by the…

A:

Q: 19. Convert the integral S(x2 + y?)dz dx dy in cylindrical coordinates. Do not evaluate the…

A: Recall: To convert from rectangular to cylindrical coordinate :       Put       x=r cosθ      y=r…

Q: 4. Evaluate the double integral D where D is enclosed by the curves y = a², y = 3x.

A: First we will draw the graph for given curves from that we will find the limits of x,y as shown…

Q: Use symmetry to evaluate the double integral. 10 sin(x) dA, R = [0, 2n] × [0, 21]

A: Integral

Q: 1 dy dx as an iterated integral. x°y Evaluate the improper integral

A: ∫8∞∫e-x11x3ydy dx=∫8∞1x3[ln y]e-x1dx = ∫8∞1x3[ln1 - lne-x]dx{ ln1 = 0 }, {ln e-x= -x.1}= ∫8∞1x3[0 -…

Q: Use Green's Theorem to evaluate the line integral , 6y² dx + 5x² dy, where Cis the boundary of the…

A: Green's Theorem: ∮C(Pdx+Qdy)=∫∫R∂Q∂x-∂P∂ydA

Q: Evaluate the integral by making an appropriate change of variables. 7 Cos (3 (V=X)) d y + x 637 6…

A:

Q: - SxS Write an integral for the area of the surface generated by revolving the curve y = cos (3x)…

A:

Q: Evaluate the integral: L x2 + y² dy dx by using polar coordinates.

A:

Q: Apply Green’s Theorem to evaluate the integrals ∮(3y dx + 2x dy)C: The boundary of 0≤ x ≤π, 0 ≤ y ≤…

A: According to Green's theorem - ∮Pdx+Qdy = ∬D dQdx-dPdydxdy          ...............(1)Given integral…