Find $\int_4^{10} {(x - 4){\rm{d}}x}$ .

Part of the graph of $f(x) = \sqrt {{x^{}} - 4}$ , for $x \ge 4$ , is shown below. The shaded region R is enclosed by the graph of $f$ , the line $x = 10$ , and the x-axis.

The region R is rotated ${360^ \circ }$ about the x-axis. Find the volume of the solid formed.

correct integration     A1A1

e.g. $\frac{{{x^2}}}{2} - 4x$, $\left[ {\frac{{{x^2}}}{2} - 4x} \right]_4^{10}$, $\frac{{{{(x - 4)}^2}}}{2}$

Notes: In the first 2 examples, award A1 for each correct term.

In the third example, award A1 for $\frac{1}{2}$ and A1 for ${(x - 4)^2}$.

substituting limits into their integrated function and subtracting (in any order)     (M1)

e.g. $\left( {\frac{{{{10}^2}}}{2} - 4(10)} \right) - \left( {\frac{{{4^2}}}{2} - 4(4)} \right),10 - ( - 8),\frac{1}{2}({6^2} - 0)$

$\int_4^{10} {(x - 4){\rm{d}}x = 18}$     A1     N2

attempt to substitute either limits or the function into volume formula     (M1)

e.g. $\pi \int_4^{10} {{f^2}} {\rm{d}}x{\text{, }}{\int_a^b {(\sqrt {x - 4} )} ^2}{\text{, }}\pi \int_4^{10} {\sqrt {x - 4} }$

Note: Do not penalise for missing $\pi$ or dx.

correct substitution (accept absence of dx and $\pi$)     (A1)

e.g. $\pi {\int_4^{10} {(\sqrt {x - 4} )} ^2}{\text{, }}\pi \int_4^{10} {(x - 4){\rm{d}}x} {\text{, }}\int_4^{10} {(x - 4){\rm{d}}x}$

volume = $18\pi$     A1     N2

[3 marks]

Many candidates answered both parts of this question correctly. In part (b), a large number of successful candidates did not seem to notice the link between parts (a) and (b), and duplicated the work they had already done in part (a). Also in part (b), a good number of candidates squared $(x - 4)$ in their integral, rather than squaring $\sqrt {x - 4}$ , which of course prevented them from noting the link between the two parts and obtaining the correct answer.

Many candidates answered both parts of this question correctly. In part (b), a large number of successful candidates did not seem to notice the link between parts (a) and (b), and duplicated the work they had already done in part (a). Also in part (b), a good number of candidates squared $(x - 4)$ in their integral, rather than squaring $\sqrt {x - 4}$ , which of course prevented them from noting the link between the two parts and obtaining the correct answer.