IB Mathematics Applications & Interpretation Topic AHL 5.12—areas Under a Curve Onto x or y Axis. Volumes of Revolution About x and y Questionbank (HL)

IB Mathematics Applications & Interpretation Topic AHL 5.12—areas Under a Curve Onto x or y Axis. Volumes of Revolution About x and y Questionbank (HL) | RevisionDojo

Practice IB Mathematics Applications & Interpretation (AI) Topic AHL 5.12—areas Under a Curve Onto x or y Axis. Volumes of Revolution About x and y with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for AHL 5.12—areas Under a Curve Onto x or y Axis. Volumes of Revolution About x and y and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Question 1

HLPaper 2

Let $y = cos^{6} (3 x)$ .

Consider the function $h (x) = 4 - 3 sin (3 x) cos^{5} (3 x)$ , defined for $0 \leq x \leq \frac{π}{6}$ .

The region $R$ is bounded by the graph of $h$ , the $x$ -axis, the $y$ -axis, and the line $x = \frac{π}{6}$ .

Determine the derivative $\frac{d y}{d x}$ .

[3]

Using your result from the previous part, find $\int sin (3 x) cos^{5} (3 x) d x$ .

[3]

State a definite integral that represents the area of $R$ .

[2]

Determine the precise area of $R$ .

[6]

Question 2

HLPaper 1

Consider the finite region $R$ bounded by the curve $y = ln (2 x + 1) - 1$ , the $y$ -axis and the $x$ -axis.

Construct a definite integral to calculate the area of $R$ .

[4]

Determine the total area of $R$ .

[2]

Question 3

HLPaper 1

The side profile of a vase is modelled by the curve $y = g (x) = 5 - 2 x + 1$ for $0 \leq x \leq 14$ .

Determine the $x$ -coordinate of the point where the graph of $g$ intersects the $x$ -axis.

[2]

The area bounded by the curve $g$ , the $y$ -axis, and the $x$ -axis is revolved $36 0^{\circ}$ around the $x$ -axis. Calculate the volume of the resulting solid of revolution.

[3]

Question 4

HLPaper 2

Let $y = sin^{5} (2 x)$ .

Consider the function $g (x) = 3 - 2 sin^{4} (2 x) cos (2 x)$ , defined for $0 \leq x \leq \frac{π}{4}$ .

The region $S$ is bounded by the graph of $g$ , the $x$ -axis, the $y$ -axis, and the line $x = \frac{π}{4}$ .

Determine the derivative $\frac{d y}{d x}$ .

[3]

Using your result from the previous part, find $\int sin^{4} (2 x) cos (2 x) d x$ .

[3]

State a definite integral that represents the area of $S$ .

[2]

Determine the precise area of $S$ .

[6]

Question 5

HLPaper 1

Lengths in this problem are measured in centimetres. Consider the function $r (x) = 1 + x$ , defined for $0 ⩽ x ⩽ 4$ . A candle is formed by rotating the region between the curve $y = r (x)$ and the $x$ -axis through $36 0^{\circ}$ about the $x$ -axis.

Calculate the total volume of the candle.

[3]

As the candle burns, the volume $W$ cm $^{3}$ of wax remaining after $t$ hours is modeled by $W = \frac{68 π}{3} e^{- 0.25 t}$ . Determine the time elapsed until half the candle remains.

[3]

IB Mathematics Applications & Interpretation AHL 5.12—areas Under a Curve Onto x or y Axis. Volumes of Revolution About x and y Questionbank

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IB Mathematics Applications & Interpretation AHL 5.12—areas Under a Curve Onto x or y Axis. Volumes of Revolution About x and y Questionbank

IB Mathematics Applications & Interpretation (AI) Topic AHL 5.12—areas Under a Curve Onto x or y Axis. Volumes of Revolution About x and y Questionbank

Unlock all 26 questions with a free account