Lesson: Line Integrals in the Plane

In this lesson, we will learn how to solve the line integral of a 2-variable function along a parameterized curve in the plane.

Worksheet: Line Integrals in the Plane • 11 Questions

Q1:

Evaluate 𝑥 + 𝑦 𝑥 + 2 𝑥 𝑦 𝑦 𝐶 2 2 d d , where 𝐶 𝑥 = 𝑡 , 𝑦 = 2 𝑡 : , 0 𝑡 1 , and 𝑡 = 𝑢 s i n for 0 𝑢 𝜋 2 .

Q2:

Evaluate 𝑥 + 𝑦 𝑥 + 2 𝑥 𝑦 𝑦 𝐶 2 2 d d , where 𝐶 𝑥 = 𝑡 , 𝑦 = 𝑡 : c o s s i n and 0 𝑡 𝜋 .

Q3:

Evaluate 𝑥 + 𝑦 𝑥 + 2 𝑥 𝑦 𝑦 𝐶 2 2 d d , where 𝐶 𝑥 = 𝑡 , 𝑦 = 2 𝑡 : 2 and 0 𝑡 1 .

Q4:

Evaluate 𝑥 + 𝑦 𝑥 + 2 𝑥 𝑦 𝑦 d d , where 𝐶 𝑥 = 𝑡 , 𝑦 = 2 𝑡 : and 0 𝑡 1 .

Q5:

Evaluate 𝑥 + 𝑦 𝑥 + 2 𝑥 𝑦 𝑦 𝐶 2 2 d d , where 𝐶 is the polygonal path from ( 0 , 0 ) to ( 0 , 2 ) to ( 1 , 2 ) .

Q6:

Calculate 𝑓 ( 𝑥 , 𝑦 ) 𝑠 𝐶 d for the function 𝑓 ( 𝑥 , 𝑦 ) and curve 𝐶 , where 𝑓 ( 𝑥 , 𝑦 ) = 2 𝑥 + 𝑦 and 𝐶 is the polygonal path from ( 0 , 0 ) to ( 3 , 0 ) to ( 3 , 2 ) .

Q7:

Calculate 𝑓 ( 𝑥 , 𝑦 ) 𝑠 𝐶 d for the function 𝑓 ( 𝑥 , 𝑦 ) and curve 𝐶 , where 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 + 𝑦 2 and 𝐶 is the path from ( 2 , 0 ) counterclockwise along the circle 𝑥 + 𝑦 = 4 2 2 to the point ( 2 , 0 ) and then back to ( 2 , 0 ) along the 𝑥 -axis.

Q8:

Calculate 𝑓 ( 𝑥 , 𝑦 ) 𝑠 d for the function 𝑓 ( 𝑥 , 𝑦 ) and curve 𝐶 , where 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 𝑥 + 1 , 𝐶 𝑥 = 𝑡 , 𝑦 = 0 : , and 0 𝑡 1 .

Q9:

Suppose that F is the gradient of the function 𝑓 ( 𝑥 , 𝑦 ) = 2 𝑥 𝑦 , and we are given points 𝑃 ( 0 , 0 ) , 𝑄 ( 1 , 0 ) , 𝑅 ( 0 , 1 ) , 𝑆 ( 1 , 1 ) , and 𝑇 ( 1 , 1 ) . Choose a starting and an end point from this set so as to maximize the integral F r d , where 𝐶 is the line between your chosen points.

Q10:

In the figure, the curve 𝐶 from 𝑃 to 𝑄 consists of two quarter-unit circles, one with centre (1, 0) and the other with centre (3, 0). Calculate the line integral 𝐶 F r d , where F = 𝑥 2 𝑦 2 s i n s i n i j 𝑥 2 𝑦 2 c o s c o s .

Q11:

Calculate 𝑓 ( 𝑥 , 𝑦 ) 𝑠 𝐶 d for the function 𝑓 ( 𝑥 , 𝑦 ) and curve 𝐶 , where 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 𝑦 , 𝐶 𝑥 = 𝑡 : c o s , 𝑦 = 𝑡 s i n , and 0 𝑡 𝜋 2 .

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