Find the line integral along the path C shown in the figure on the right \int_C (x^2+y^2)dy 12110601481.jpg

BolkowN

BolkowN

Answered question

2021-05-03

Find the line integral along the path C shown in the figure on the right
C(x2+y2)dy

Answer & Explanation

tabuordg

tabuordg

Skilled2021-05-04Added 99 answers

The solution is written in the photo:

image
image

Jeffrey Jordon

Jeffrey Jordon

Expert2021-09-08Added 2605 answers

Consider the integral

0(x2+y2)dy

The objective is to find the line integral along the path C. From the graph the line integeral split into two parts,

C(x2+y2)dy=C1(x2+y2)dy+C2(x2+y2)dy

Here, C is the path along the curve

x2+y2 from (0,0) to (2,0) to (2,3)

Here, C1 is the path along the curve from (0,0) to (2,0) and C2 is the path along the curve from (2,0) to (2,3)

The equation of the line passing throuhgh the points (0,0) to (2,0)

x020=y000, y=0

That implies, dy=0

Therefore, the integral is,

C1(x2+y2)dy=C1(x2+y2)(0)=0

The equation of the line passing through the points (2,0) to (2,3)

x222=y030

x=2

Therefore, the integral is,

C1(x2+y2)dy=03(22+y2)dy

=03(4+y2)dy

=(4(y)+(y33))03

=(4(3))+(33)

=21

Therefore, the cine integral along path C is,

C(x2+y2)dy=C1(x2+y2)dy+C2(x2+y2)dy

=0+21

=21

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