Use Green's Theorem to evaluate F * dr. (Check the orientation of the curve before applying the theorem.)F(x, y) = sqrtx+ 4y^3, 4x^2 + sqrt(y)C consists of the arc of the curve y = sin(x) from (0, 0) to (pi, 0) and the line segment from (pi, 0) to (0, 0)

Mylo O'Moore

Mylo O'Moore

Answered question

2021-01-13

Use Green's Theorem to evaluate Fdr. (Check the orientation of the curve before applying the theorem.)
F(x,y)=x+4y3,4x2+y
C consists of the arc of the curve y=sin(x) from (0, 0) to (π,0) and the line segment from (π,0) to (0, 0)

Answer & Explanation

unessodopunsep

unessodopunsep

Skilled2021-01-14Added 105 answers

Step 1
The given function is,
F(x,y)=(x+4y3,4x2+y)
Step 2
C is a closed curve and using Green’s theorem for clockwise orientation the integral is evaluated using the below formula.
CFdr=D(QxPy)dA
Where, F=(P,Q)=(x+4y3,4x2+y)
Step 3
Calculate the partial derivative of P and Q as follows.
Qx=x(4x2+y)
=8x
Py=y(x+4y3)
=12y2
Step 4
Evaluate the integral Fdr by substituting the limits as follows.
CFdr=0π0sinx(8x+12y2)dydx
=0π(8xy+12y33)0sinxdx
=0π(8x(sinx)+4sin3x)dx
=0π8xsinxdx+0π4sin3xdx
=8(x(cosx)cosxdx)0π+40πsin2xsinxdx
=8(xcosxsinx)0π+40π(1cos2x)sinxdx
=8π+4(cosxcos3x3)0π
=8π+4(22

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