IB Mathematics Applications & Interpretation Topic…

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

(a) the $x$ -coordinate of the point where the graph of $g$ intersects the $x$ -axis.

(b) The area bounded by the curve $g$ , the $y$ -axis, and the $x$ -axis is revolved $360^\circ$ around the $x$ -axis. the volume of the resulting solid of revolution.

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

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

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

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

Lengths in this problem are measured in centimetres. Consider the function $r(x)=1+\sqrt{x}$ , defined for $0\leqslant x\leqslant 4$ . A candle is formed by rotating the region between the curve $y=r(x)$ and the $x$ -axis through $360^\circ$ about the $x$ -axis.

(a) the total volume of the candle.

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

The sketch shows part of the curve $x=4-e^{2y}$ , where $y\in\mathbb{R}$ . Region $S$ is shaded, enclosed by the curve, the $x$ -axis and the $y$ -axis. $shaded region enclosed by the curve x=4-e^{2y} and the coordinate axes$

(a) Express the area of region $S$ as a definite integral.

Lengths in this problem are measured in centimetres. Consider the function $h(x)=2+\sqrt{x}$ , defined for $0\leqslant x\leqslant 9$ . A scented candle is formed by rotating the region between the curve $y=h(x)$ and the $x$ -axis through $360^\circ$ about the $x$ -axis.

(b) As the candle burns, the volume $W$ cm $^3$ of wax remaining after $t$ hours is modeled by $W=\frac{297\pi}{2}e^{-0.2t}$ . the time elapsed until half the candle remains.

(a) the total volume of the candle.

(a) the derivative $\frac{dy}{dx}$ .

(d) the precise area of $S$ .

(b) the total area of region $S$ .

(c) Consider a triangle with vertices at $P(0,0)$ , $Q(2,1)$ and $R(0,k)$ . the value of $k$ such that the area of this triangle is equal to the area of region $S$ .

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 AHL 5.12—areas Under a Curve Onto x or y Axis. Volumes of Revolution About x and y Questionbank

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 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