1.35: Past-paper question 7

Figure 1 shows a sketch of part of the curve $\displaystyle C$ with equation

The point $\displaystyle P$ lies on $\displaystyle C$ and has coordinates $\displaystyle (4, a)$

(a) Show that $\displaystyle a = 6$

The line $\displaystyle L$ is the normal to $\displaystyle C$ at the point $\displaystyle P$

(b) Show that an equation of $\displaystyle L$ is $\displaystyle 5y + 4x - 46 = 0$

The finite region $\displaystyle R$ is bounded by the curve $\displaystyle C$, the line $\displaystyle L$, the $\displaystyle x$-axis and the line with equation $\displaystyle x = 1$

(c) Use calculus to find the exact area of $\displaystyle R$