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

One App. One Place for Learning.

Most popular questions from this chapter

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