4. the value of the given integrals | z - i | = 2,(z € C) around the circle compute for positive defined C curve 1 dz z² + 4 b) 1 dz (z² + 4)²

Q: 2. Give iterated integrals for (xz*)dS where S is the part of the cylinder x² + y = 9 above the…

A: Given that x2+y2=9 and the part of the cylinder which is above XY plane implying that z > 0. We…

Q: 4/ Find the area of the region enclosed between the curves y= cos 2x, y=0, x=TT/4, x= TT/2 ? ans/…

A:

Q: verify Green's Theorem by evaluating both integrals * dx + x*dy = aM dA for the given path.

A:

Q: Choose five other iterated integrals that are equal to the given iterated integral. x² f(x, y, z) dz…

A:

Q: Calculate the integral of f(æ, y) = 10x over the region D bounded above by y = x(2 – x) and below by…

A:

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

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

Q: Evaluate the line integral, where C is the given cu curve. (x + yz) dx + 2x dy + xyz dz, C consists…

A: C consist line from A (2, 0, 1) to B(3, 3, 1) Now,

Q: ON Verify Green's Theorem by evaluating both integrals 7y dx + 7x*dy = J J]a дм ƏA for the path ду…

A:

Q: Verify Green's Theorem by evaluating both integrals ƏN y² = / / dx + x2 dy dA - Əx ду for the given…

A:

Q: 2. Consider the region bounded by the graphs of y = 3* – 1 and x = y² – 3, illustrated at the right.…

A: The objective is to setup the integral that calculate the center of mass of the region bounded by…

Q: Evaluate the line integral, where C is the given curve. (x + 8y) dx + x2 dy, C consists of line…

A:

Q: Evaluate the double integral. (x2 + 2y) dA, D is bounded by y = x, y = x³, x > 0

A:

Q: #6. Evaluate the line integral (y dx +z dy +x dz), where C is given by x = Vt, y = t, z = t? with…

A:

Q: Verify Green's Theorem by evaluating both integrals aN dx + x² dy = / / dA ay y2 ax for the given…

A: To verify green's theorem by evaluating both the integrals.

Q: up an integral for the length of x = cos y , 0<x <; using ds = () dx. Set 1+ dx, A. ds = [2 1+ 1-x2…

A:

Q: 1. Give iterated integrals for r (x² + yz)dV where E is in the first octant bounded by the planes: y…

A:

Q: Evaluate the double integral. // (x2 + 2y) dA, Dis bounded by y = x, y = x³, x > o

A: To evaluate the below double integral ∫∫Dx2+2ydA ,   D is bounded by y=x, y=x3,x≥0

Q: Verify Green's Theorem by evaluating both integrals aM dA ƏN | y? dx + x2 dy = ду, дх for the given…

A:

Q: 2. Set up an integral to find the exact length of the curve defined by 1 f(r) 1 In r %3D where 1<r <…

A:

Q: Evaluate the line integral, where C is the given curve. (x + 9y) dx + x² dy, C consists of line…

A: To evaluate the line integral, consider the curveC1 consisting of the line segment from0,0to9,1. The…

Q: Compute the following integral: ez -dz z(z – 3) where y = {z : |z| = 2} positively oriented.

A:

Q: *0 = x = 0, y = 1, x = y about the line y = 1 can be computed using the method of disks or washers…

A:

Q: Define f : [0, 2] → R by f(x) = 1 for 0 sx S 1 and f(x) = 2 for 1 < x< 2. Show that fE R(x) on [0,…

A: Consider the given function.

Q: 4. Compute the integral x sin (xy) dA where R is the region 0 <x <1 and 0<y < .

A:

Q: Evaluate the double integral. [[ ×20 (2+2y) dA, D is bounded by y = x, y = 2³₁x20

A: off

Q: Evaluate the iterated integral. 27 1²th I 729- y dx dy = 27 1.² 1. 729-y² y dx dy

A:

Q: 3) Show that the line integral is independent of path and then evaluate the inte S, sin y dx + (x…

A:

Q: Determine the integral of x y dx dy. Use inner limits=0 to (y + 1) and outer limits=0 to 1.

A:

Q: Calculate the tripe integrals SSS xyz dxdydz where B é the block 0≤x≤2; 0 ≤ y ≤ 1; 1≤z≤2

A: We have given in question, triple integration of xyz with respect to x, y, z

Q: Use Green's Theorem to evaluate the line integral of C to the function (y − x) dx + (2x − y) dy  for…

A: To evaluate: ∫Cy-xdx+2x-ydy using Green's theorem where C is the boundary of the region lying…

Q: 1. For F(x, y) = (y – x, 2x – y) over the region D bounded by x = y2 – y and y = x, verify Green's…

A: according to the problem, we have

Q: 1. Sketch the solid of integration of the given integral. Evaluate the integral. M. a? dV; S is…

A:

Q: Calculate the integral Sy3dx + (x3 + 3xyl)dy in the following ways; A) in shape direct. B) using…

A:

Q: 4. Use Green's Theorem to evaluate the line integral: f ye dx + x*y² dy, where C is the from (-2,0)…

A:

Q: Evaluate the triple integral. III. 7z dv, where E is bounded by the cylinder y2 + z² = 9 and the…

A:

Q: Compute the double integral SS s cat+zy) clxdy R where Ris the aren boun d eel by y--V, y:-x 一<メ4

A: The region of integration is between the given lines where the blue curve is the function y=-x and…

Q: 2. Show that, in the region D: 0 < x<1, 0 <y< x², the following double integral 1 dA =- In(2) x5 + 1…

A:

Q: 1. Evaluate the double integral /xcos ydA, where D is bounded by y = 0, y =x, and x = 3, Answer:

A: Evaluate: ∫∫DxcosydA Where D bounded by y=0, y=x2 and x=3 When y=0, x=0

Q: Sketch the region R bounded below by the curve y = x², on the left by the line x = 1 and above by…

A: Consider the given integral is∫∫Rx2+2xydydx and R is the region bounded below by y=x2 on the left of…

Q: You are rotating the region bounded by y 2 = x, y = 1, and x = 0 about the line x = 1. Set up the…

A: First, we sketch the region using a graphing calculator  y2=x , y=1 , x=0 The shaded region is…

Q: Evaluate the double integral. 7x cos(y) dA, D is bounded by y = 0, y = x2, x = 7 %3D

A:

Q: Use Green's Theorem to evaluate the line integral. S y² dx + xy dy C: boundary of the region lying…

A:

Q: 3. Approximate the integral using: (a) (b) 11 trapezoidal rule. Simpson's rule. 0.8 [8 dx

A:

Q: Evaluate the triple integral. III. 5z dv, where E is bounded by the cylinder y2 + z2 = 9 and the…

A:

Q: Compute the integral ∮C [(cos x − 3y) dx + (2x − sin y) dy], where C is the closed curve that…

A: The region of integration C is divided into three line segments C1, C2 and C3. Let C1:y=0,…

Q: Evaluate the double integral. (x2 + 2y) dA, D is bounded by y = x, y = x³, x 2 0

A: Consider the following graph:

Q: Verify Green's Theorem by evaluating both integrals ƏN [< y² dx + x² dy = √₂₁ / (™N - 3W) CIA dA əx…

A:

Q: Suppose that f(c, y): at which {(x, y) | 0 s< 1, - a <yS vE}. 1+ x Then the double integral of f(x,…

A: According to question given that f(x,y)=y1+x at which {x,y) |0≤x≤1, -x≤y≤x

Q: 8) consider the region R in the first quadrant(xz0, y2) bounded by x=2y² X=Y²+1 a) set up, but clo…

A:

Q: 4/ Find the area of the region enclosed between the curves y= cos 2x , y=0, x=TT/4, x= TT/2 ? ans/…

A: Given-y=cos(2x)y=0x=π4x=π2Let consider that area of the curve is =AWe have two value of x and two…

Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: