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Problem 45

In Exercises $43-46,$ use a CAS to perform the fo…

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Problem 44

In Exercises $43-46,$ use a CAS to perform the following steps to evaluate the line integrals.
a. Find $d s=|\mathbf{v}(t)| d t$ for the path $\mathbf{r}(t)=g(t) \mathbf{i}+h(t) \mathbf{j}+$ $k(t) \mathbf{k} .$
b. Express the integrand $f(g(t), h(t), k(t))|\mathbf{v}(t)|$ as a function of the parameter $t .$
c. Evaluate $\int_{C} f d s$ using Equation $(2)$ in the text.
$$f(x, y, z)=\sqrt{1+x^{3}+5 y^{3}} ; \quad \mathbf{r}(t)=t \mathbf{i}+\frac{1}{3} t^{2} \mathbf{j}+\sqrt{t} \mathbf{k} 0 \leq t \leq 2$$

Answer

a) $d s=\sqrt{1+\frac{4 t^{2}}{9}+\frac{1}{4 t}} d t$
b) $f(g(t), h(t), k(t))|v(t)|=\left(\sqrt{1+t^{3}+\frac{5}{27} t^{6}}\right)\left(\sqrt{1+\frac{4 t^{2}}{9}+\frac{1}{4 t}}\right)$
c) $\int_{C} f \cdot d s \approx 5.8080$



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Video Transcript

Okay. What we want to do is we want to, um right or find, um we went to find d s, which is equal to V of tea. Um, de ti four, um, r of t is equal to t I plus t square J plus three t squared. Okay, um and so we know that the, uh t is equal to the derivative, which is gonna be a plus to TJ plus six t k. And so, um, the magnitude is gonna be the square root of one squared plus two t squared plus 60 squared, which is gonna give me the square root of one plus 40 t squared. And so D s is equal to the square root of one plus 40 t squared TT. Okay. And then now we want to express, um, the in a grand of our function. And our function is X Y Z is equal to the square root of one plus 30 x squared plus 10. Why and t goes from 0 to 2 inclusive. And so my inner role is going to be from 0 to 2. Ah, my function in terms of tea. And so we know exes. T so I would have a 30 t squared there plus a And I know why is T squares or a 10 T squared? So this would be a one plus 40 t square times d s, which is a one plus 40 t squared TT. So here is my integral. And now what we want to do is we want to go ahead and use a, um either a online, um, calculator or your calculator to actually evaluate this integral. And so I'm gonna go ahead, um, and pull up. I haven't 84 plus emulator. And so what we're gonna do is we're going to evaluate that integral using the calculator. So on mine, I go to math and I go all the way down here to function integration. And I'm going from zero 22 and I have the square root. But I noticed Now, couple things we can knows that we have a square root of one place for a t square times another one, which means, um, this is just going to be a one plus a 40. And unfortunately, I'm gonna have to use X squared in here. Um, and then I'm integrating. Um, not with respect to Esso, with respect to ask X and hit enter. So I get 108.67 So this is actually equal to, um 100 and eight 0.67

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