QUESTION $1$

$($a$)$ Find the area below $f(x)=-x^2+4x+3$ and above $g(x)=-x^3+7x^2-10x+5$ over the interval $1x2$

$(15$ marks$)$

$($b$)$ A $1$ cm radius circular bacterial colony on an agar plate has a certain density, highest at the middle and drops off toward the edge.

The density can be described by the equation $(r)=1-r^2$ in units of one million cells per square centimetres. What is the total number of bacteria in the colony?

It might be helpful to know that to calculate this, you'll need to calculate the integral:

$N_{\text{t}\text{o}\text{t}\text{a}\text{l}}={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}dA,$ where $A$ is the appropriate area element. All answers are given in millions of cells.

$(10$ marks$)$