9.5 Integrating Vector-Valued Functions

In AP Calculus, you've learned about various techniques for evaluating integrals of real-valued functions, such as substitution, integration by parts, and partial fractions. These methods can also be extended to integrate parametric and vector-valued functions, which are functions that have multiple variables or output multiple values.

Parametric functions are defined by two or more real-valued functions that describe the position of a point in space as a function of some parameter, such as time or angle. To integrate a parametric function, students can use techniques such as change of variables and substitution to express the integral in terms of the parameter and then evaluate the definite or indefinite integral.

Vector-valued functions are defined by multiple real-valued functions that output a vector, such as a position or velocity vector. To integrate a vector-valued function, students can use techniques such as line integrals and Green's theorem to evaluate the integral of the vector-valued function.

Recall from previous sections that vector-valued functions are functions that output multiple values, often in the form of a vector. These functions can be used to model physical phenomena such as position, velocity, and acceleration, and can be represented in Cartesian or polar coordinates.

To take the integral of a vector-valued function, we can break it down into its individual x- and y-components and integrate each component separately. This can be done using techniques such as substitution, integration by parts, and partial fractions, which are also used to integrate real-valued functions. The result will be a vector-valued function, which gives the displacement of the particle or the velocity or acceleration at a particular point in time.

One common application of vector-valued function integration is to calculate displacement, which is the change in position of a particle over a given interval of time. By taking the definite integral of the velocity vector-valued function with respect to time, we can find the displacement vector, which gives the position of the particle at the end of the interval.

Another application is to move backwards along the position-velocity-acceleration chain, by taking the indefinite integral of the acceleration vector-valued function with respect to time. This gives us the velocity vector-valued function, then by taking the indefinite integral of the velocity vector-valued function gives us the position vector-valued function. 🚆