Two curves, ${{C}_{1}}$ and ${{C}_{2}}$, have the equation $y={{x}^{2}}-10x+26$ and $y=-{{x}^{2}}+2a

Question

Two curves, ${{C}_{1}}$ and ${{C}_{2}}$, have the equation $y={{x}^{2}}-10x+26$ and $y=-{{x}^{2}}+2ax+\left( 5-{{a}^{2}} \right)$ respectively, where $a$ is a constant. For some value of $a$, the two curves will intersect (at two distinct points) and form a bounded region. Let $A$ be the area within this bounded region from $x=4$ to $x=6$. Show that the value of $a$ that maximises $A$ is 5. Verify it is indeed maximum. [5] Using this value of $a$, find the exact volume of revolution of $A$ about the $x$-axis. [2]

Accepted Answer

(i) $a = 5$ (ii) $40\pi$ units³

Maximum Area Bounded Between Two Curves

Maximum Area Bounded Between Two Curves

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