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Chapter 14: Q. 24 (page 1107)

In Exercises 21鈥�28, evaluate the multivariate line integral of the given function over the specified curve.

f(x,y)=x2+y2, with C the unit circle traversed clockwise.

Short Answer

Expert verified

The multivariate line integral of the given function over the specified curve is 鈭�Cf(x,y)ds=2蟺.

Step by step solution

01

Step 1. Given Information

In the given exercises we have to evaluate the multivariate line integral of the given function over the specified curve.

f(x,y)=x2+y2, with C the unit circle traversed clockwise.

02

Step 2. The given function is f(x,y)=x2+y2

So the line integral is

鈭�Cf(x,y)ds=鈭�abf(r(t))(x'(t))2+(y'(t))2dt

Parametrization: x=cos(t),y=-sin(t),0鈮�t鈮�2蟺.So,dx=-sin(t)dt,dy=-cost.

Now findingf(r(t))

f(r(t))=x2+y2f(r(t))=(cost)2+(-sint)2 f(r(t))=1

03

Step 3. Now solving the ∫Cf(x,y)ds=∫abf(r(t))(x'(t))2+(y'(t))2dt

鈭�Cf(x,y)ds=鈭�02蟺1(cost)2+(-sint)2dt鈭�Cf(x,y)ds=鈭�02蟺1cos2t+sin2tdt鈭�Cf(x,y)ds=鈭�02蟺11dt鈭�Cf(x,y)ds=鈭�02蟺dt鈭�Cf(x,y)ds=t02蟺鈭�Cf(x,y)ds=2蟺-0鈭�Cf(x,y)ds=2蟺

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Most popular questions from this chapter

Problem Zero: Read the section and make your own summary of the material.

Find the areas of the given surfaces in Exercises 21鈥�26.

S is the portion of the surface parametrized by r(u,v)=(3u-v,v+u,v-u) whose preimage (the domain in the uv-plane) is the unit square [0,1]脳[0,1]

Q. True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: Stokes鈥� Theorem asserts that the flux of a vector field through a smooth surface with a smooth boundary is equal to the line integral of this field about the boundary of the surface.

(b) True or False: Stokes鈥� Theorem can be interpreted as a generalization of Green鈥檚 Theorem.

(c) True or False: Stokes鈥� Theorem applies only to conservative vector fields.

(d) True or False: Stokes鈥� Theorem is always used as a way to evaluate difficult surface integrals.

(e) True or False: Stokes鈥� Theorem can be interpreted as a generalization of the Fundamental Theorem of Line Integrals.

(f) True or False: If F(x, y ,z) is a conservative vector field, then Stokes鈥� Theorem and Theorem 14.12 together give an alternative proof of the Fundamental Theorem of Line Integrals for simple closed curves.

(g) True or False: Stokes鈥� Theorem can be interpreted as a generalization of the Fundamental Theorem of Calculus.

(h) True or False: Stokes鈥� Theorem can be used to evaluate surface area .

Find the work done by the vector field

F(x,y)=xex2+y2x2+y2i+yex2+y2x2+y2j

in moving an object around the unit circle, starting and ending at (1,0).

Give a formula for a normal vector to the surface S determined by y = g(x,z), where g(x,z) is a function with continuous partial derivatives.
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