Applications of Integration
Samuel Baker
11 min read
Study Guide Overview
This study guide covers integration applications, including calculating the average value of a function, modeling particle motion (position, velocity, acceleration, displacement, distance traveled) and net change, and determining areas and volumes. Key concepts include the average value formula, definite integrals for displacement and distance, accumulation functions, finding areas between curves (with intersections), calculating volumes using cross-sections (square, rectangular), the disc method, and the washer method. Additionally, for BC Calculus, arc length and distance traveled are covered.
In this unit, you’ll learn how to find the average value of a function, model particle motion and net change, and determine areas and volumes. Specifically, you should develop an understanding of integration that can be transferred across many other applications. You'll start to see how integration is useful in the fields of physics and engineering, and exercise your drawing skills as you draw sketches to visualize 2D and 3D shapes. With integrals, you won’t have to use any of the Riemann sums you learned earlier in the year. It is crucial to understand the general steps for solving each problem in this unit. This unit should be about 10-15% of the AP Calculus AB Exam or 6-9% of the AP Calculus BC Exam.
#8.1 Finding the Average Value of a Function on an Interval
In calculus, the average value of a function is the value that the function would take at a single point if the area under the curve were equal to the area of a rectangle with the same width and height as the curve. In other words, it is a way to measure the "center of mass" or "balance point" of a function.
The average value of a function f(x) over the interval [a,b] is given by the formula:
To find the average value of a function, you can first set up the definite integral for the function over the interval of interest. Then, you can evaluate the definite integral using the appropriate techniques, such as substitution.
#8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals
If you have taken or are taking a physics course, then you already know what position, velocity, and acceleration are. If you haven't heard these terms before, you only need to know that position is where an object is at a moment in time, velocity is the rate of change (read as derivative) of the position as a function of time, and acceleration is the rate of change (read as derivative) of velocity as a function of time.
Definite integrals can be used to calculate the displacement and total distance traveled for a particle in linear motion over a certain interval of time.
Displacement is a vector quantity that represents the change in position of an object. It can be found by taking the definite integral of the velocity function with respect to time. This is because velocity is the rate of change of position, and the definite integral of a rate of change is the change itself.
On the other hand, the total distance traveled is a scalar quantity that represents the total distance covered by the object, regardless of its final position. It can be found by taking the definite integral of the speed function with respect to time. This is because speed is the magnitude of the velocity and the definite integral of a scalar function represents the accumulated change in the function over a certain interval.
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