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 the curve $y = 3 \sin 2 x + 6 \sin x$ and its maximum point $M$ . Find the exact area of the shaded region.

Accepted Answer

Given that, we have,

Given that, the curve is $y = 3 \sin 2 x + 6 \sin x$

Differentiate $y$ with respect to $x$ , then we get: $\frac{d y}{d x} = 3 \cdot 2 \cdot \cos 2 x + 6 \cos x$

$\Rightarrow \frac{d y}{d x} = 6 \cos 2 x + 6 \cos x$

For maxima or minima, we have, $\frac{d y}{d x} = 6 \cos 2 x + 6 \cos x = 0$

$\Rightarrow 6 (\cos 2 x + \cos x) = 0$

$\Rightarrow \cos 2 x + \cos x = 0$

$\Rightarrow 2 \cos^{2} x - 1 + \cos x = 0$

$\Rightarrow 2 \cos^{2} x + 2 \cos x - \cos x - 1 = 0$

$\Rightarrow 2 \cos x (\cos x + 1) - 1 (\cos x + 1) = 0$

$\Rightarrow (2 \cos x - 1) (\cos x + 1) = 0$

$\Rightarrow (2 \cos x - 1) = 0 or (\cos x + 1) = 0$

$\Rightarrow \cos x = \frac{1}{2} or \cos x = - 1$

$\Rightarrow x = n π \pm \frac{π}{3} or x = (2 n + 1) π$ , where $n \in Z$

$\Rightarrow x = \frac{π}{3}$ [Since, the angle lies in first quadrant, so neglect all other values]

Thus, the maximum point $x = \frac{π}{3}$ .

Therefore, required area $A = \int_{0}^{\frac{π}{3}} y d x = \int_{0}^{\frac{π}{3}} (3 \sin 2 x + 6 \sin x) d x$

$\Rightarrow A = {(- \frac{3}{2} \cos 2 x - 6 \cos x)}_{0}^{\frac{π}{3}}$

$\Rightarrow A = (- \frac{3}{2} \cos 2 (\frac{π}{3}) - 6 \cos (\frac{π}{3})) - (- \frac{3}{2} \cos 2 (0) - 6 \cos (0))$

$\Rightarrow A = (- \frac{3}{2} (- \frac{1}{2}) - 6 (\frac{1}{2})) - (- \frac{3}{2} - 6)$

$\Rightarrow A = (\frac{3}{4} - 3) - (- \frac{15}{2})$

$\Rightarrow A = - \frac{9}{4} + \frac{15}{2}$

$\Rightarrow A = \frac{21}{4}$

Hence, the exact area of the shaded region is $\frac{21}{4}$ sq. units.

The diagram shows the curve $y = 3 \sin 2 x + 6 \sin x$ and its maximum point $M$ . Find the exact area of the shaded region.

Important Questions on Integration

Learn and Prepare for any exam you want !

The diagram shows the curve y=3sin2x+6sinx and its maximum point M. Find the exact area of the shaded region.

Important Questions on Integration

Learn and Prepare for any exam you want !

The diagram shows the curve $y = 3 \sin 2 x + 6 \sin x$ and its maximum point $M$ . Find the exact area of the shaded region.