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In Exercises $43 - 46 ,$ use a CAS to perform the following steps to evaluate the line integrals.

$$

\begin{array} { l } { \text { a. Find } d s = | \mathbf { v } ( t ) | d t \text { for the path } \mathbf { r } ( t ) = g ( t ) \mathbf { i } + h ( t ) \mathbf { j } + k ( t ) \mathbf { k } \text { . } } \\ { \text { b. Express the integrand } f ( g ( t ) , h ( t ) , k ( t ) ) | \mathbf { v } ( t ) | \text { as a function of the parameter } t . } \\ { \text { c. Evaluate } \int _ { C } f d s \text { using Equation } ( 2 ) \text { in the text. } } \end{array}

$$

$$

\begin{array} { l } { f ( x , y , z ) = \sqrt { 1 + 30 x ^ { 2 } + 10 y } ; \quad \mathbf { r } ( t ) = t \mathbf { i } + t ^ { 2 } \mathbf { j } + 3 t ^ { 2 } \mathbf { k } } \\ { 0 \leq t \leq 2 } \end{array}

$$

A. $d s=\left(\sqrt{1+40 t^{2}}\right) d t$

B. $f(g(t), h(t), k(t))|v(t)|=\left(\sqrt{1+40 t^{2}}\right)\left(\sqrt{1+40 t^{2}}\right)$

C. $\int_{C} f \cdot d s \approx 108.67$

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Johns Hopkins University

Harvey Mudd College

University of Nottingham

Idaho State University

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

University of Central Arkansas