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Evaluate the line integral $\int_{C} \mathbf{F} \cdot d \mathbf{r},$ where $C$ is given by the vector function $\mathbf{r}(t) .$

$$\mathbf{F}(x, y)=x y \mathbf{i}+3 y^{2} \mathbf{j}$$

$$\mathbf{r}(t)=11 t^{4} \mathbf{i}+t^{3} \mathbf{j}, \quad 0 \leqslant t \leqslant 1$$

45

Vector Calculus

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So this question we're asked to the German delighted to grow of f doctor your over the cat along the path to see were given that f of x com ally is X y. I have plus three wife's where jihad and we're told that our tea is equal to 11 t to the power for I have musty Q J at steam ranges from 0 to 1. All right, so, um, weaken right back Kovacs Why this convenient way as X by common three wise Where and then when we look at our position vector uh, we realize that axes 11 to you the powerful. And why is Cube so again? What? We're every tragedy to the German half of our tea. We're just gonna plug in our values of X and y that are now in terms of tea. So, for example, X Y is going to be 11 t to the power four times. Uh, cute. Uh, common. Now again, why is t cubes? So we're gonna put that in for wise of three d Cube square. So we get that after our tea is 11 t to the power of seven comma three t to the power six. All right, now we want to determine our prime. So our prime iscause we want everything in terms of so we just take the derivatives of the derivative of eleventy The power forward respected t is just 40 40 Cube and the derivative of Tea Cube with respect to t is just a three d square. Okay, So again, our task is to determine the light in general. And so another way to write decided to grow. And the interdependence is integral from a to B of ethical or tea. Not our private T DT. So this, uh so this is in terms of tea. So we found f of our tea right here. We determined our primal to you right here. And our our limits are from 01 because they said that t ranges from Syria one. So, um but don't product we simply have promoted by this first term with this first term. Plus, this term kinds this trip. So 11 times 44 is 484 and to the power of seven times to Cubist E to the power of 10 plus now three t to the power of six times three t to the power to is 90 to the power of a DT. So if we take the integral of this, um so the integral of Tito Power 10 is just t to the power Love in divided by 11 and the integral of T to the power of eight is just t to the power of nine, divided by nine. So we get this right over here. 484 divided by 11 is just 44. And yes, so this is 34 and then now we just plug in the limits of the integral. So it's 44 times one to the power of 11. Must work to the power of nine, which is 45. There were begins 0 40 we get she's 44 times zero to the power of 11 is just zero and plus zero to the power of +90 So we get zero. So 45 +90 is 45. So the value far lining to grill is 40