 Hello, and welcome to this screencast on section 9.7, derivatives and integrals of vector-valued functions. This screencast is going to cover integrating a vector-valued function and projectile motion. An antiderivative of a vector-valued function, little r, is a vector-valued function, capital R, such that if we take the derivative of capital R, we get little r. The indefinite integral of a vector-valued function, r, is the general antiderivative of r and represents the collection of all the antiderivatives of r. These definitions should look familiar from single-variable calculus. Also, recall that the general antiderivative includes an added constant, usually denoted by c, in order to indicate that the general antiderivative is in fact an entire family of functions. The same reasoning that allowed us to differentiate a vector-valued function in the last video, component-wise, applies to integrating as well. If we wish to compute the integral of a vector-valued function, r, we can take the integral of its x component, the integral of its y component, and the integral of its z component, separately, and sum these three integrals. In light of being able to integrate and differentiate component-wise with vector-valued functions, we can solve many problems that are analogous to those we encountered in single-variable calculus. Anytime that an object is launched into the air with a given velocity and launch angle, the path the object travels is determined almost exclusively by the force of gravity. Whether in sports such as archery or shot put, in military applications with artillery, or in important fields like firefighting, it's important to be able to know when and where a launched projectile will land. Assume that we fire a projectile from a launcher and the only force acting on the fired object is the force of gravity pulling down on the object. That is, we assume no effect due to spin, wind, or air resistance. With these assumptions, the motion of the object will be planar, so we can assume that the motion occurs in two-dimensional space. If an object is launched from an initial position, x-naught, y-naught, with an initial velocity, v-naught, at an angle, theta, with the horizontal, then the position of the object at time t is given by the following vector-valued function r. This assumes that the only force acting on the object is the acceleration g due to gravity.