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I am currently working on the homework for 16.2. Number one, for example, states: "Compute $\int\limits_Cxy^2ds$ along the line segment from (1,2,0) to (2,1,3)".

I am setting it up as follows:

For my parameterized line I get: $$\vec{p}(t) = \langle1,2,0\rangle + t\langle1,-1,3\rangle$$

Giving me: $$x=t+1$$ $$ x' = 1$$ $$y=2-t$$ $$ y'= -1$$ $$z=3t$$ $$ z'= 3$$

Then setting up an integral to compute:

$$\int(t+1)(2-t)^2\sqrt{(1)^2+(-1)^2+(3)^2}dt$$

What I am not sure of is how to know what my bounds of integration are. I would love any help on this or to know if what I have done so far is correct or not correct...Thanks!

I am currently working on the homework for 16.2. Number one, for example, states: "Compute $\int\limits_Cxy^2ds$ along the line segment from (1,2,0) to (2,1,3)".

I am setting it up as follows:

For my parameterized line I get: $$\vec{p}(t) = \langle1,2,0\rangle + t\langle1,-1,3\rangle$$

Giving me: $$x=t+1$$ $$ x' = 1$$ $$y=2-t$$ $$ y'= -1$$ $$z=3t$$ $$ z'= 3$$

Then setting up an integral to compute:

$$\int(t+1)(2-t)^2\sqrt{(1)^2+(-1)^2+(3)^2}dt$$

What I am not sure of is how to know what my bounds of integration are. Do I plug my point values in for x, y, and z? Or in this case, for x and y and then solve for t? Or just for x?

I would love any help on this or to know if what I have done so far is correct or not correct...Thanks!