#Accumulation Functions

#Table of Contents

1. What is an Accumulation Function?
2. Writing Accumulation Functions as Definite Integrals
3. Worked Example
4. Practice Questions
5. Glossary
6. Summary and Key Takeaways

#What is an Accumulation Function?

For example, if a strawberry harvester gathers half a kilogram of strawberries for each meter of strawberry plants (rate of change = 0.5 kilograms per meter), the accumulation function from the harvester's starting point to a point $x$ meters from that starting point is simply $0.5x$. Substituting in a value for $x$ gives you the amount of strawberries harvested up to that point.

#Writing Accumulation Functions as Definite Integrals

Most often, you will see accumulation functions written as definite integrals:

$$g(x) = \int_{a}^{x} f(t) , dt$$

• $g$: the accumulation function
  • $g$ is a function of $x$
  • Its value changes as the value of $x$ changes
• $f$: the associated rate of change function
• $a$: the (fixed) starting point of the integral
• $x$: the (variable) ending point of the integral
• $t$: a 'dummy variable' used for evaluating the integral
  • Any letter except $x$ can be used for the dummy variable inside the integral

A definite integral is being used to define a new function of $x$.

#Worked Example

Let $f$ be the function defined by $f(x) = 2x + \sin x$.

Let $g$ be the function defined by $g(x) = \int_{0}^{x} f(t) , dt$.

#(a) Show that $g(x) = x^2 - \cos x + 1$.

Answer:

We need to integrate $f(t)$:

$$\begin{align*} \int_{0}^{x} f(t) , dt &= \int_{0}^{x} (2t + \sin t) , dt \\ &= \left[ t^2 - \cos t \right]_{0}^{x} \\ &= \left( x^2 - \cos x \right) - \left( 0^2 - \cos 0 \right) \\ &= x^2 - \cos x - (0 - 1) \\ &= x^2 - \cos x + 1 \end{align*}$$

Therefore, $g(x) = x^2 - \cos x + 1$.

#(b) Let $g(x)$...

#Practice Questions

Practice Question

1. Define an accumulation function for a car traveling at a constant speed of 60 km/h.
2. Given $f(x) = 3x^2 + \cos x$, find the accumulation function $g(x)$ from $0$ to $x$.
3. If $h(x) = \int_{2}^{x} (4t + \sin t) , dt$, calculate $h(5)$.
4. Explain why a dummy variable is necessary in defining an accumulation function.

#Glossary

• Accumulation Function: A function that represents the accumulation of changes over an interval.
• Definite Integral: An integral with upper and lower limits, providing an accumulation function.
• Rate of Change: The rate at which one quantity changes with respect to another.
• Dummy Variable: A placeholder variable used in the context of integration.

#Summary and Key Takeaways

• Accumulation functions help us understand the total change over an interval.
• They are often written using definite integrals.
• The integral's limits define the starting and ending points of the accumulation.
• The rate of change function is integrated to find the accumulation function.
• Always use a dummy variable inside the integral to avoid confusion with the variable of integration.