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Chapter 6: Q11P (page 314)

Evaluate each of the following integrals in the easiest way you can.

鈭�C(xsinx-y)dx+(x-y2)dy,where Cis the triangle in the (x, y) plane with

vertices (0,0), (1,1), and (2,0).

Short Answer

Expert verified

The solution to this problem is I=2

Step by step solution

01

Given Information

The given information is鈭�Cxsinx-ydx+x-y2dy.

02

Definition of Green’s Theorem

The Green's theorem connects a line integral around a simple closed curve C to a double integral over the plane region D circumscribed by C in vector calculus. Stokes' theorem has a two-dimensional special case.

03

Find the solution

Write the values of P and Q.

P=xsinx-yQ=x-y2

Find the difference of their differentiation

鈭�Q鈭�x-鈭�P鈭�y=1+1=2

The integral is in the form mentioned below.

I=鈭�dydx=2脳area鈥�of鈥�the鈥�triangle=2

Hence, the solution to this problem is .I=2

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Most popular questions from this chapter

鈭�(2xi鈭�2yj+5k)脳苍诲蟽over the surface of a sphere of radius 2and center at the origin.

Given 蠒=x2-yz and the point (3,4,1) find

(a) 鈭�蠁 at P ;

(b) a unit vector normal to the surface 蠁=5 at P ;

(c) a vector in the direction of most rapid increase of 蠁 at P;

(d) the magnitude of the vector in (c);

(e) the derivative of at in a direction parallel to the liner=i-j+2k+(6i-j-4k)t.

鈭�F脳苍诲蟽where F=(y2鈭�x2)i+(2xy鈭�y)j+3zkand 蟽is the entire surface of the tin can bounded by the cylinder

role="math" localid="1657353627256" x2+y2=16

role="math" localid="1657353639412" z=3

role="math" localid="1657353647648" z=-3

Do case (b) of Example 1above.

Let F = i - 5j + 2kact at the point (2, 1, 0) Find the torque of F about the line -(3j + 4k)- 2it .
See all solutions

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