2 (2y-y² 15. Evaluate the integral / x² + y²dzdydx by using cylindirical coordinates.

Q: Determine the value of the integral Integral sign: ((z^2 + r^2)^-3/2)x((2r)dr)

A:

Q: -4-y Evaluate triple integral , S dz dy dx .

A:

Q: Q2: Evaluate the following triple integral 1 c1 1. , S S,(x² + y² + z²)dzdydx 2. S" S y sinz dxdydz…

A:

Q: Find a formula for the integral. dy = y 3(1– 6y) Incorrect

A: We have ∫e6xe3ydy

Q: Show your complete solution: Evaluate the triple integral pl+y² xdtdydx 1-x S S T 2y

A:

Q: raluate the double integral R 3ædA where Ris the triangle with vertices (-1,0), (0,1) and (1,0).

A:

Q: Evaluate the line integral -ydr + rdy along the curve C : y² = 6x from (6, 6) to (0,0). NOTE: Enter…

A: The given line integral is ∫c-ydx+xdy along the curve c: y2=6x from (6,6) to (0,0)

Q: Evaluate the triple iterated integral. 2/2 L7 xyz dx dy dz 0 Jo J-1

A:

Q: Calculate the double integral. 4x dA, R = [0, 4] × [0, 1] 1 + xy

A: The given integral is ∬R 4x1+xy dA  where R=0,4×0,1. Evaluate as follows.

Q: Use symmetry to evaluate the double integral. 4xy 1+ x2 dA, R= {(x, Y) | -4 s x s 4,0 s y s 1}

A: When the upper and lower limits of definite integrals become same, the result is zero irrespective…

Q: Evaluate the line integral -ydx + xdy along the curve C: = = 21x from (21, 21) to (0,0). %3D NOTE:…

A:

Q: Evaluate the iterated integral. V 16 - 16 – y2 | (x + y) dx dy Submit Apcwor

A: To find the following integralI=∫04∫016-y2x+ydx.dy

Q: 2 (2y-y² a 15. Evaluate the integral / / |zVx² + y²dzdrdy by using cylindirical coordinates. 0 0

A:

Q: Evaluate the line integral [21r²yzds with respect to s along the 2 curve C : x = t₁ y = t², z =…

A:

Q: +4)(v² +®y)Vy? +8y – 1 dy 3.

A:

Q: Use symmetry to evaluate the double integral. 2xy J63 JR 1 + x ,10 dA, R = {(x, y) | -4 ≤ x ≤ 4,0 ≤…

A: Consider the given double integral, ∫∫R2xy1+x10dA, where R=x,y-4≤x≤4, 0≤y≤1. Consider the integrand,…

Q: Show that the form under the integral sign is exact in the plane and •(2,2) evaluate the integral /…

A: Given the line integral is ∫-1,-12,2e-x2-y2xdx+ydy. The line integral∫CF·dr=∫CF1dx+F2dy.

Q: 70. Solid bounded above by x² + y² + z² = 36 and below by = 2 Vx + y?

A: Given: Consider The volume of any solid E in R3

Q: dy Evaluate the integral| 2. y - 14y + 65

A: We first complete the square in the denominator and apply the formula:

Q: Evaluate the double integral. (2x + y) dA, D = {(x, y) | 1 <ys 2, y – 1 < x< 1}

A:

Q: 00 1 (b) Calculate the improper integral dx , or show that it (x² + 25)3/2 -00 diverges . Show your…

A:

Q: Two models, R, = 7.21+0.62t and R, = 7.21+0.47t, are given for revenue (in billions of dollars per…

A:

Q: Evaluate the integral 4 (y 2)(2y1) dy Jo

A: To evaluate,

Q: Use the property to estimate the best possible bounds of the integral. II. 5 sin“(x + y) dA, T is…

A: To estimate the best bounds of the integral.

Q: S (y + 3x)dydx Evaluate:

A:

Q: Evaluate the iterated integral. (r – y) dydx 2x

A: The given integral is ∫04∫2x2x-ydydx.

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

A:

Q: Evaluate the line integral |-ydx + xdy along the curve C : y² = 24x from (24, 24) to (0, 0).

A:

Q: Evaluate the triple integral (show all your work by hand!) |T| 2.xyz dzdydx.

A:

Q: 2 V2y-y2 a 15. Evaluate the integral x² + y²dzdxdy by using cylindirical coordinates. 0.

A:

Q: Estimate the double integral_R  (3xy^(2)+2x^(2)) dA where R= [0,1] x [0,1] by diving R into four…

A: We need to evaluate integral.

Q: ST * /sin(x+y)dxdy (b) Evaluate the double integral J0 π/8, π/4, 3 π /8 , π/2 by using trapezoidal…

A: If a two-dimensional domain Ω can be decomposed into rectangles, then the integral of a function…

Q: evaluate the triple iterated integral.

A:

Q: -dr evaluate the integrals by substitution method J 2dx V2+r /2+x

A:

Q: Solve the problem using convolution integral   y′′ + 2y′ + 2y = f(t), y(0) = 0, y′(0) = 0

A: Convolution Integral: Let f(t) and g(t) be defined in [0,∞). Then, the convolution of f(t) and g(t),…

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

A:

Q: Evaluate the triple integral: 2 x (Iny (z) dzdydx

A: ∫12∫1x∫y2ln(y)zdzdydx=∫12∫1xz22y2ln(y)dydx=12∫12∫1xlny2-y4dydx Now,…

Q: The surface area of the solid of revolution generated by rotating 3 x = (y² + 2)ž,1 sys2 about the…

A: Given function is x=13y2+332 where 1≤y≤2. The surface area of revolution of the function is given by…

Q: Q2/(a): Evaluate the double integral , 3xy dA over the rectangle dA = dydx} R {(x, y): -3 x< 2,…

A:

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

A:

Q: 2y-ya Evaluate the integral 12 + y?dzdydr by using cylindirical coordinates.

A: Let z = z , x = r cos(theta) , y = r sin(theta).

Q: 2 V2y- 15. Evaluate the integral v zVr+ ydzdydr by using cylindirical coordinates.

A: As per the company policy i can answer only first question for you,more than that leads me towards…

Q: Evaluate the iterated integral. 3 4 5 /// + x² + z² ) dx dy dz -3 0 0 3 4 5 !! (x² + y² + z² )dx dy…

A: We have to find

Q: late the double integral. (6z²y – 5y*) dA, R= {(z,y) | 0< z S 3,0 syS1}

A: Given: The double integral ∫∫R6x2y3-5y4dA,       R=x, y| 0≤x≤3, 0≤y≤1. Here, y varies from 0 to 1…

Q: Evaluate the integral. V3 9earctan(y) dy 1 + y2

A:

Q: 2 V2y-y2 Evaluate the integral x²+ y²dzdydx by using cylindirical coordinates.

A:

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. inegral 1  to 2 [(4 y2 -7y -12 )/ y (y+2)(y-3) ]dy

A: Given:                  ∫124y2-7y-12y(y+2)(y-3)dy   To determine:                    We can solve…

Q: Evaluate the iterated integral. 5 2 y dy dx 10 Vx+y 5 2

A: Given, I=∫15∫02 yx+y2dy dx To Evaluate I.

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: