1.26: Tangent, Normal, Area and Volume

The curve $\displaystyle C$ has equation

The point $\displaystyle P$ lies on $\displaystyle C$ and has $\displaystyle x$ coordinate $\displaystyle 4$.

The line $\displaystyle k$ is the tangent to $\displaystyle C$ at $\displaystyle P$.

(a) Find an equation for $\displaystyle k$, giving your answer in the form

where $\displaystyle a$, $\displaystyle b$ and $\displaystyle c$ are integers.

The line $\displaystyle l$ is the normal to $\displaystyle C$ at $\displaystyle P$.

(b) Find an equation for $\displaystyle l$, giving your answer in the form

where $\displaystyle d$, $\displaystyle e$ and $\displaystyle f$ are integers.

(c) Find the area of the triangle bounded by the line $\displaystyle k$, the line $\displaystyle l$ and the $\displaystyle x$-axis.

The finite region bounded by $\displaystyle C$, the line $\displaystyle l$, the $\displaystyle x$-axis and the $\displaystyle y$-axis is rotated through $\displaystyle 360^\circ$ about the $\displaystyle x$-axis.

(d) Use algebraic integration to find, to the nearest whole number, the volume of the solid generated.