   Chapter 16, Problem 8RE

Chapter
Section
Textbook Problem

Evaluate the line integral.8. ∫C F · dr, where F(x, y) = xy i + x2 j and C is given by r(t) = sin t i + (1 + t) j, 0 ⩽t ⩽ π

To determine

To Evaluate: The line integral CFdr , where F(x,y)=xyi+x2j and C is given by r(t)=sinti+(1+t)j , 0tπ .

Explanation

Given data:

F(x,y)=xyi+x2j , r(t)=sinti+(1+t)j , 0tπ .

Formula used:

Write the expression for r(t) in terms of x and y .

r(t)=xi+yj (1)

Write the required differential formulae to evaluate the given integral.

ddxxn=nxn1ddxcosx=sinxddxsinx=cosx

Write the required integration formulae to evaluate the given integral.

[f(x)]ndx=[f(x)]n+1n+1

Write the standard formula.

sinxcosx=12sin2x

Consider the expression for r(t) as follows.

r(t)=sinti+(1+t)j (2)

Compare equations (1) and (2).

x=sinty=(1+t)

Consider the expression for F(x,y) as follows.

F(x,y)=xyi+x2j

Substitute sint for x and (1+t) for y ,

F(x,y)=sint(1+t)i+sin2tj

Differentiate equation (2) with respect to t .

dr=(costi+j)dt

Find the value of CFdr .

CFdr=C(sint(1+t)i+sin2tj)(costi+j)dt

Change the limits with respect to t

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