How do you evaluate the following line integral #(x^2)zds#, where c is the line segment from the point (0, 6, 1) to the point (4,1,5)?
Here's the integral calculation:
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To evaluate the line integral ( \int_C x^2 z , ds ), where ( C ) is the line segment from the point ( (0, 6, 1) ) to the point ( (4, 1, 5) ), follow these steps:

Parametrize the curve ( C ) with a vectorvalued function ( \mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle ) where ( t ) varies from ( t = t_1 ) to ( t = t_2 ), and ( \mathbf{r}(t_1) ) corresponds to the initial point and ( \mathbf{r}(t_2) ) corresponds to the final point on the curve.

Calculate the derivative of the parametric equations to find ( ds ).

Substitute the parametric equations for ( x ), ( y ), and ( z ) into the integrand ( x^2 z ) to express it in terms of ( t ).

Compute the integral ( \int_{t_1}^{t_2} (x(t))^2 z(t) , ds ) using the limits ( t = t_1 ) to ( t = t_2 ).
Let's proceed with the solution:

Parametrize the curve ( C ): [ \mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle = \langle 4t, 6  5t, 1 + 6t \rangle ] where ( 0 \leq t \leq 1 ).

Calculate the derivative of ( \mathbf{r}(t) ) to find ( ds ): [ \mathbf{r}'(t) = \sqrt{(4)^2 + (5)^2 + (6)^2} = \sqrt{16 + 25 + 36} = \sqrt{77} ] [ ds = \mathbf{r}'(t) , dt = \sqrt{77} , dt ]

Substitute the parametric equations into the integrand: [ x(t) = 4t, \quad y(t) = 6  5t, \quad z(t) = 1 + 6t ] [ (x(t))^2 z(t) = (4t)^2 (1 + 6t) = 16t^2 (1 + 6t) ]

Compute the integral: [ \int_{0}^{1} 16t^2 (1 + 6t) \sqrt{77} , dt ]
This integral can be evaluated using standard techniques, such as integration by parts or usubstitution.
After integrating, you will obtain the value of the line integral.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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