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AustraliaVIC
VCE 12 Methods 2023

7.11 Applications of integration

Interactive practice questions

In a closed habitat, the population of kangaroos $P\left(t\right)$P(t) is known to increase according to the function $P'\left(t\right)=\frac{t}{2}+9$P(t)=t2+9, where $t$t is measured in months since counting began.

a

Determine the total change in the population of kangaroos in the first $4$4 months since counting began.

b

Solve for $T$T, the number of months it will take from when counting began for the population of kangaroos to increase by $88$88.

Easy
8min

The population growth rate of fish is given by $p\left(x\right)$p(x), as shown in the graph below.

The total population growth of fish over $n$n years is given by $\int_0^np\left(x\right)dx$n0p(x)dx as shown in the table below.

Easy
3min

An object is cooling and its rate of change of temperature, after $t$t minutes, is given by $T'=-10e^{-\frac{t}{5}}$T=10et5.

Easy
6min

Calculate the average rate of change $f\left(x\right)$f(x) if $f'\left(x\right)=12\sin6x$f(x)=12sin6x over the domain $\frac{\pi}{6}\le x\le\frac{\pi}{3}$π6xπ3.

Easy
5min
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Outcomes

U34.AoS3.10

application of integration to problems involving finding a function from a known rate of change given a boundary condition, calculation of the area of a region under a curve and simple cases of areas between curves, average value of a function and other situations.

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