Name: Differentiation: The Chain Rule
Uploaded: 2019-07-19
Duration: 7 min 13 s
Description: In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions, f and g, in terms of f and g.

Question

The diagram shows part of the curve $y = \cos x + 2 \sin x$ . Find the exact volume of the solid formed when the shaded region is rotated through $360 °$ about the $x -$ axis.

Accepted Answer

Given that, we have,

Given that, the curve is $y = \cos x + 2 \sin x$

When the shaded area is rotated $360 °$ about $x -$ axis, a volume is generated.

So the volume $V$ of the solid of revolution is given by the integral of volume expression over the limits, $V = \int_{a}^{b} π y^{2} d x$

Inserting values given in the problem we obtain, $V = \int_{0}^{\frac{π}{2}} π {(\cos x + 2 \sin x)}^{2} d x$

$\Rightarrow V = π \int_{0}^{\frac{π}{2}} (\cos^{2} x + 4 \sin^{2} x + 4 \sin x \cos x) d x$

$\Rightarrow V = π \int_{0}^{\frac{π}{2}} (\frac{1 + \cos 2 x}{2} + 4 (\frac{1 - \cos 2 x}{2}) + 2 \sin 2 x) d x$

$\Rightarrow V = π \{{[\frac{1}{2} x + \frac{\sin 2 x}{4} + 2 x - 2 \times \frac{\sin 2 x}{2} - 2 \times \frac{\cos 2 x}{2}]}_{0}^{\frac{π}{2}}\}$

$\Rightarrow V = π \{{[\frac{1}{2} x + \frac{\sin 2 x}{4} + 2 x - \sin 2 x - \cos 2 x]}_{0}^{\frac{π}{2}}\}$

$\Rightarrow V = π \{[\frac{1}{2} \times \frac{π}{2} + \frac{\sin 2 (\frac{π}{2})}{4} + 2 (\frac{π}{2}) - \sin 2 (\frac{π}{2}) - \cos 2 (\frac{π}{2})] - [\frac{1}{2} \times 0 + \frac{\sin 2 (0)}{4} + 2 (0) - \sin 2 (0) - \cos 2 (0)]\}$

$\Rightarrow V = π \{[\frac{π}{4} + π + 1] - [- 1]\}$

$\Rightarrow V = π [\frac{5 π}{4} + 2]$

Hence, the exact volume of solid is $π [\frac{5 π}{4} + 2] {cm}^{3}$ .

The diagram shows part of the curve $y = \cos x + 2 \sin x$ . Find the exact volume of the solid formed when the shaded region is rotated through $360 °$ about the $x -$ axis.

Important Questions on Integration

Learn and Prepare for any exam you want !

The diagram shows part of the curve y=cosx+2sinx. Find the exact volume of the solid formed when the shaded region is rotated through 360° about the x-axis.

Important Questions on Integration

Learn and Prepare for any exam you want !

The diagram shows part of the curve $y = \cos x + 2 \sin x$ . Find the exact volume of the solid formed when the shaded region is rotated through $360 °$ about the $x -$ axis.