Chapter 11: Q. 10 (page 871)

$State what it means for a vector function r (t) = <x (t), y (t), z (t)> to be integrable .$

Short Answer

Expert verified

$\int r (t) dt = i \int x (t) dt + j \int y (t) dt + k \int z (t) dt is an antiderivative of the function <x (t), y (t), z (t)>$

Step by step solution

Step 1. Given information is

$r (t) = <x (t), y (t), z (t)>$

Step 2. Definition of integration of vector function

$Let r (t) = <x (t), y (t), z (t)> where x = x (t), y = y (t), z = z (t) be three real valued functions . Let \int x (t) dt, \int y (t) dt and \int z (t) dt be the antiderivatives of x (t), y (t) and z (t) respectively on I \subseteq R . Then the vector function : \int r (t) dt = \int (x (t) i + y (t) j + z (t) k) dt \int r (t) dt = i \int x (t) dt + j \int y (t) dt + k \int z (t) dt is an antiderivative of the function <x (t), y (t), z (t)> For this, \int_{a}^{b} r (t) dt = i \int_{a}^{b} x (t) dt + j \int_{a}^{b} y (t) dt + k \int_{a}^{b} z (t) dt$

Step 3.Example

$Let r (t) = <- cost, sint, t^{3}> \int r (t) dt = \int <- cost, sint, t^{3}> dt \int r (t) dt = i \int - cost dt + j \int sint dt + k \int t^{3} dt \int r (t) dt = i (sint + c_{1)} + (- cost + c_{2}) + k (\frac{t^{4}}{4} + c_{3}) where c_{1}, c_{2} and c_{3} are scalar constants . = <- sint, - cost, \frac{t^{4}}{4}> + c where c is vector constant .$

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$State what it means for a vector function r (t) = <x (t), y (t), z (t)> to be integrable .$

Short Answer

Step by step solution

Step 1. Given information is

Step 2. Definition of integration of vector function

Step 3.Example

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Statewhatitmeansforavectorfunctionr(t)=x(t),y(t),z(t)tobeintegrable.

Short Answer

Step by step solution

Step 1. Given information is

Step 2. Definition of integration of vector function

Step 3.Example

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Most popular questions from this chapter

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$State what it means for a vector function r (t) = <x (t), y (t), z (t)> to be integrable .$