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Problem 5 Easy Difficulty

Evaluate the line integral, where $C$ is the given curve.
$$\begin{array}{l}{\int_{c}\left(x^{2} y^{3}-\sqrt{x}\right) d y,} \\ {C \text { is the arc of the curve } y=\sqrt{x} \text { from }(1,1) \text { to }(4,2)}\end{array}$$

Answer

$\int_{C}\left(x^{2} y^{3}-\sqrt{x}\right) d y=\frac{243}{8}$

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Video Transcript

All right, so in this question, were asked to determine finding to grow of X square times like U minus Swiffer to vex. And we're given that the integral that the curve goes from, why equals Discriminative X is the Yes, this is the length of the Ark of Wise equals Discriminative x 1112 for two. So first thing we can do is we can rewrite X as why square. So we just square both sides. We get that X is equal to y squared. All right, great. So now if we try to change everything in our original into groups in terms off lie, maybe you would be able to integrate this. So the first thing we do, wherever there is an ex, we can put my square. Okay, So instead of extraditing football, I swear. But then this is X squared. So it's why Squared square, we have, like, cubed minus the square root of X is just what But what we notice is we converted everything in terms of and then we know that our curve are why values go for 1 to 2, and that inside here we have everything in terms of What? So why did the power for times like you distrust white it a seven minus y And then now this is just a regular integral which we know how to solve because everything's in terms off. Why? So the integral off Why did the power of seven is why? To the power of a divided by eight minus. Now the integral applies just why squared, divided by two. I remember our wives both of 1 to 2. So now if we plug this back in, so this is a definite integral. So what we do is first we put into Lebanon, we subtract Sadr and then we put it one. All right, So now, due to the power of eight is 256 and then to swear it is just for so. These are two values minus one to the power of eight is just one and one squared is just And then now what we can do is we can multiply by for over four here because we want a common denominator. So four divided by four and the same thing he we're gonna multiply by four divided. All right, so now we have our all our denominators are eight. And then if we do, the algebra is 156 minus 16 minus four minus four. They get 243 divided.

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