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Chapter 14: Q 11. (page 1154)

Evaluate the line integral of the given function over the specified curve.

鈭�CF(x,y).drwhere F(x,y)=7y2i-3xyjand C is the line segment x=t,y=3tfor0鈮�t鈮�1

Short Answer

Expert verified

鈭�CF(x,y).dr=12

Step by step solution

01

Given Information

The given expression is F(x,y)=7y2i-3xyj,x=t,y=3tlocalid="1652874571194" x=t,y=3t.

02

To evaluate line integral

The curve is parametrized as,

r(t)=t,3tfor 0鈮�t鈮�1

Differentiating it with respect to t,

r'(t)=1,3

Finding field vector by using values of x,y,

Fxt,yt=73t2i-3t3tj

=63t2i-9t2j

=63t2,-9t2

Line integral is calculated as,

鈭�CF(x,y).dr=鈭�CF(x(t),y(t)).r'(t)dt

=鈭�C63t2,-9t2路(1,3)dt

localid="1652328520867" =鈭�0163t2路1+-9t2路3dt(Simplify dot product)

localid="1652956410638" =鈭�0163t2-27t2dt=鈭�0136t2dt

=12t301

=121-120

=12

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