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Integration of Vector Valued Function
12
Used in Physics as well:
Applications:
The motion of a projectile:
If an object is launched from a point
with initial velocity at an angle with the
horizontal, then the position of the
object at time t is given by:
• The integrals of vector-valued functions are very useful for engineers,
physicists, and other people who deal with concepts like force, work,
momentum, velocity, and movement. For instance, the velocity of an
object can be described as the integral of the vector-valued function
that describes the object's acceleration.
Integration of Vector-valued Function
Conclusion:
DONE
• Integrating a vector-valued function involves finding the anti derivative of each component of the function and combining them to obtain the overall anti derivative. This process is useful in calculating displacement vectors, work done by force fields, and fluid flow over a given path or region. The specific
application determines the ultimate significance of the integration .
WORKING
TO DO
IN PROGRESS
Definition
• Integration of vector valued functions involves finding the antiderivative of
each component of the vector function and combining them to form the
antiderivative of the vector function. This process is similar to integrating
scalar functions, but each component of the vector function is treated
separately .
• The result of integration will be a new vector-valued function, or, if you
compute a definite integral, a new vector. So, if F → ( t ) = ( f ( t ) , g ( t ) )
and F → ( t ) is a vector-valued function, then ∫ F → ( t ) = ( ∫ f ( t ) d t ,
∫ g ( t ) d t ) .
Presented By:
Ilsa Faisal
Umer Farooq
Umair Sarwar
Abdul Moiz
Example:
Representation:
Let's say we have the vector function F(t) = cos(t), sin(t),t
To integrate this function, we need to find the
antiderivative of each component of the vector function.
∫cos(t) dt = sin(t) + C1
∫sin(t) dt = -cos(t) + C2
∫t dt = 1/2 t^2 + C3
Combining these antiderivatives, we get the
antiderivative of the vector function:
∫F(t) dt = <sin(t), -cos(t), 1/2 t^2> + C
So the antiderivative of the vector function F(t) is sin(t), -
cos(t), 1/2 t^2 plus a constant of integration C.
• Integration of a vector-valued function involves finding the integral of each
component function separately. A vector-valued function is typically represented
as:
r(t) = f(t), g(t), h(t)
where f(t), g(t), and h(t) are scalar functions of the parameter t.
• To integrate the vector-valued function, you need to integrate each component
function with respect to t:
∫r(t) dt = ∫f(t) dt, ∫g(t) dt, ∫h(t) dt
• The result will be a new vector-valued function, where each component is the
integral of the corresponding component in the original function.
References
References
- https://math.libretexts.org/
- https://chatgpt.ai/
- https://www.ck12.org/calculus/indefinite-integral-of-a-vector-valued- function/lesson/integration-of-vector - valued-functions-calc/