Q 81. Consider the function f(x)=x聽co... [FREE SOLUTION]

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Chapter 5: Q 81. (page 430)

Consider the function $f (x) = x \cos 2 x$ .
(a) Find the signed area between the graph of $f (x)$ and the $x$ -axis on $[0, \frac{3 蟺}{4}]$ shown in the figure next at the left.
(b) Find the area between the graph of $f (x)$ and the graph ofrole="math" localid="1650063304032" $g (x) = - \cos 2 x$ on $[0, \frac{3 蟺}{4}]$ shown in the figure next at the right.

Short Answer

Expert verified

Part (a). The required area is $- 1.4275$ .

Part (b). The required area is $- 1.9275$ .

Step by step solution

Part (a) Step 1. Given information.

The given function is $f (x) = x \cos 2 x$ and $g (x) = - \cos 2 x$ .

Part (a) Step 2. Find the area.

The required area is:

$\begin{array}{rcl} {鈭�}_{0}^{\frac{3 蟺}{4}} (x \cos 2 x) d x & = & {[x 鈭� \cos 2 x d x - 鈭� \{\frac{d}{d x} (x) 鈭� \cos 2 x d x\} d x]}_{0}^{\frac{3 蟺}{4}} \\ = & {[x 路 \frac{\sin 2 x}{2} - 鈭� \frac{\sin 2 x}{2} d x]}_{0}^{\frac{3 蟺}{4}} \\ = & {[x 路 \frac{s i n 2 x}{2} + \frac{1}{4} \cos 2 x]}_{0}^{\frac{3 蟺}{4}} \\ = & [\frac{3 蟺}{4} 路 \frac{s i n (\frac{3 蟺}{2})}{2} + \frac{1}{4} \cos (\frac{3 蟺}{2}) - 0 - \frac{1}{4} \cos (0)] \\ = & \frac{1}{2} 路 \frac{3 蟺}{4} (- 1) - \frac{1}{4} \\ = & - \frac{3 蟺}{8} - \frac{1}{4} \\ = & - 1.4275 \end{array}$

Hence, the required area is $A_{1} = - 1.4275$ .

Part (b) Step 1. Given information.

The given function is $f (x) = x \cos 2 x$ and $g (x) = - \cos 2 x$ .

Part (b) Step 2. First, find the definite integral ∫03π4-cos2x dx.

$\begin{array}{rcl} A_{2} & = & {鈭�}_{0}^{\frac{3 蟺}{4}} - \cos 2 x d x \\ = & {[\frac{- \sin 2 x}{2}]}_{0}^{\frac{3 蟺}{4}} \\ = & [\frac{- \sin (\frac{3 蟺}{2})}{2} - 0] \\ = & \frac{1}{2} \\ = & 0.5 \end{array}$

Part (b) Step 3. Find the area between the graph of f(x) and g(x).

$\begin{array}{rcl} A & = & {鈭�}_{0}^{\frac{3 蟺}{4}} f (x) - {鈭�}_{0}^{\frac{3 蟺}{4}} g (x) \\ = & A_{1} - A_{2} \\ = & - 1.4275 - 0.5 \\ = & - 1.9275 \end{array}$

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Short Answer

Step by step solution

Part (a) Step 1. Given information.

Part (a) Step 2. Find the area.

Part (b) Step 1. Given information.

Part (b) Step 2. First, find the definite integral ∫03π4-cos2x dx.

Part (b) Step 3. Find the area between the graph of f(x) and g(x).

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Consider the function f(x)=xcos2x.(a) Find the signed area between the graph of f(x)and the x-axis on 0,3蟺4shown in the figure next at the left.(b) Find the area between the graph off(x)and the graph ofrole="math" localid="1650063304032" g(x)=-cos2xon0,3蟺4shown in the figure next at the right.

Short Answer

Step by step solution

Part (a) Step 1. Given information.

Part (a) Step 2. Find the area.

Part (b) Step 1. Given information.

Part (b) Step 2. First, find the definite integral ∫03π4-cos2x dx.

Part (b) Step 3. Find the area between the graph of f(x) and g(x).

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Logic and Functions

Geometry

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Calculus

Study anywhere. Anytime. Across all devices.