20N.1.SL.TZ0.S_3

pestleMathematicsAISLPaper 120N· ahl-5-12-areas-under-a-curve-onto-x-or-y-axis-volumes-of-revolution-about-x-and-ysource ↗

Let $f (x) = \sqrt{12 - 2 x}, x \leq a$ . The following diagram shows part of the graph of $f$ .

The shaded region is enclosed by the graph of $f$ , the $x$ -axis and the $y$ -axis.

The graph of $f$ intersects the $x$ -axis at the point $(a, 0)$ .

Find the value of $a$ .

[2]

Find the volume of the solid formed when the shaded region is revolved $360 °$ about the $x$ -axis.

[5]

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

recognize $f (x) = 0$ (M1)

eg $\sqrt{12 - 2 x} = 0, 2 x = 12$

$a = 6$ (accept $x = 6$ , $(6, 0)$ ) A1 N2

[2 marks]

attempt to substitute either their limits or the function into volume formula (must involve $f^{2}$ ) (M1)

eg $\int_{0}^{6} f^{2} d x, π \int {(\sqrt{12 - 2 x})}^{2}, π \int_{0}^{6} 12 - 2 x d x$

correct integration of each term A1 A1

eg $12 x - x^{2}, 12 x - x^{2} + c, {[12 x - x^{2}]}_{0}^{6}$

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

eg $π (12 (6) - {(6)}^{2}) - π (0), 72 π - 36 π, (12 (6) - {(6)}^{2}) - (0)$

Note: Award M0 if candidate has substituted into $f, f^{2}$ or $f'$ .

volume $= 36 π$ A1 N2

[5 marks]

Examiners’ report

[N/A]