Review question

# What's the area between $f_k(x) = x(x-k)(x-2)$ and the $x$-axis? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8130

## Suggestion

Let $0< k <2$. Below is sketched a graph of $y=f_k(x)$ where $f_k(x) = x(x-k)(x-2)$. Let $A(k)$ denote the area of the shaded region.

1. Without evaluating them, write down an expression for $A(k)$ in terms of two integrals.

How is the value of an integral affected if a function gives negative values (is below the $x$-axis)?

1. Explain why $A(k)$ is a polynomial in $k$ of degree $4$ or less. [You are not required to calculate $A(k)$ explicitly.]

What is the highest of power of $k$ we can get by integrating $f$, with the limits given?

1. Verify that $f_k (1+t) = -f_{2-k}(1-t)$ for any $t$.
1. How can the graph of $y=f_k(x)$ be transformed to the graph of $y = f_{2-k}(x)$?

What happens if we use part (iii), replacing $1+t$ with $x$?

Deduce that $A(k)=A(2-k)$.

Can we compare $f_{2-k}(x)$ to $f_k (x)$? What do the transformations from part iv) do to the area $A(k)$?

1. Explain why there are constants $a, b, c$ such that $A(k) = a(k-1)^4 + b(k-1)^2 + c.$ [You are not required to calculate $a,b,c$ explicitly.]

Can we show $A(1+k) = A(1-k)$? What implications for symmetry does this have?