A084totalchange


 * Total Change using Definite Integral **

If given a rate of change function, we can find the total change over a given period of time.

If the rate of change of the function is given by: math . \qquad \dfrac{df}{dx} \qquad. math

Then the total change between x = a and x = b is given by

math . \qquad \displaystyle{ \int\limits_a^b \; \dfrac{df}{dx} \; dx } \qquad. math


 * Example 1 **

A water tank is leaking such that the volume changes according to:

math . \qquad \dfrac{dV}{dt} = -4e^{-0.5t} \quad t \geqslant 0 \qquad. math ... ... ... where t is measured in minutes and V is measured in cm 3

Find the total amount of water that leaks from the tank in the first 6 minutes (correct to 1 decimal place).


 * Solution: **

math \text{Change in Volume } = \displaystyle{ \int\limits_0^6 \; -4e^{-0.5t} \; dt } \\. \\ . \qquad \qquad \qquad \qquad = \Big[ 8e^{-0.5t} \Big]_0^6 \\. \\ . \qquad \qquad \qquad \qquad = \Big( 8e^{-3} \Big) - \Big( 8e^0 \Big) \qquad. \\ . \\ . \qquad \qquad \qquad \qquad = -7.6 \; cm^3 math

Hence 7.6 cm 3 of water leaks from the tank.


 * Example 2 **

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