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### Evaluate $\int_{C} \sqrt{x^{2}+y^{2}} d s$ along â€¦

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University of Central Arkansas
Problem 11

# Evaluate $\int_{C}(x y+y+z) d s$ along the curve $\mathbf{r}(t)=2 t \mathbf{i}+$ $t \mathbf{j}+(2-2 t) \mathbf{k}, 0 \leq t \leq 1$

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## Video Transcript

okay. What we want to dio is we want to evaluate, um, over the curve on the line integral of X y plus, why play Z d s? Um, along the curve R t is equal to to t i plus t j plus to minus two tea. Okay. And t t goes from 0 to 1 inclusive. Okay, so the first thing we need to do is we know that d s is equal to the magnitude of e of T. Um d t. And so first thing we need to do is we know that the of t is equal to the derivative, um, of that curve r t. So this is gonna be equal to two i plus J minus two K. And so, um, the magnitude of e of t is equal to the square root of t squared plus one squared plus a native to squared. And so this is gonna give me, um three. Um, and so Gs is equal to three d t. Okay, so this is gonna be the integral from 0 to 1. Um, X is to t. Why is t and Z is T minus two t and three D t so that seeing a girl we want to do, um, so this is gonna be equal to, um, three times, integral from 0 to 1. Ah, to t squared minus t plus two tt. So this is gonna be three times 2/3 t cute, minus 1/2 t squared, plus to t evaluated, um, 0 to 1. And so when we do that, um, we should get 13 house.

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