# Suppose that in a memory experiment the rate of memorizing is given by M'(t)= -0.005t^2 + 0.5t,...

## Question:

Suppose that in a memory experiment the rate of memorizing is given by {eq}M'(t)= -0.005t^2 + 0.5t {/eq}, where M'(t) is the memory rate, in words per minute. How many words are memorized in the first 10 minutes?

## Application of Integration:

The application of integration is used to find the total rate of change of the function in the given interval. The form of the definite integral to find the total rate of change form t = a to t = b is:

{eq}\displaystyle \int_{a}^{b} f'(t) \, dt {/eq}

We have:

{eq}M'(t) = -0.005t^2 + 0.5t {/eq}

Integrating M'(t) with respect to the time t from t = 0 to t = 10:

{eq}\displaystyle \int_{0}^{10} -0.005t^2 + 0.5t \, dt \\ \displaystyle = \left [ -\frac{0.005t^3}{3} + \frac{0.5t^2}{2} \right ]_{0}^{10} \\ \displaystyle = \left [ -\frac{0.005(10)^3}{3} + \frac{0.5(10)^2}{2} \right ] \\ \displaystyle = \left [ -\frac{5}{3} + \frac{50}{2} \right ] \\ \displaystyle \approx 23.33 {/eq}

Implies that, around 23 words can be memorized in first 10 minutes.