Q. 48 For each pair of functions f聽an... [FREE SOLUTION]

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Chapter 4: Q. 48 (page 386)

For each pair of functions $f$ and $g$ and interval $[a, b]$ in Exercises $41 - 52$ , use definite integrals and the Fundamental Theorem of Calculus to find the exact area of the region between the graphs of $f$ and $g$ from $x = a t o x = b$ .
$f (x) = \sin x, g (x) = \cos x, [a, b] = [- \frac{蟺}{2}, \frac{蟺}{2}]$

Short Answer

Expert verified

The exact area is $2 .$

Step by step solution

Step 1. Given Information.

The given function and interval is $f (x) = \sin x, g (x) = \cos x, [a, b] = [- \frac{蟺}{2}, \frac{蟺}{2}]$

Step 2. Graph of the functions.

The graph of the functions will be,

Step 3. Required area.

The exact area will be,

${鈭�}_{- \frac{蟺}{2}}^{\frac{蟺}{2}} | f (x) - g (x) | d x = = {鈭�}_{- \frac{蟺}{2}}^{\frac{蟺}{4}} (g (x) - f (x)) d x + {鈭�}_{\frac{蟺}{4}}^{\frac{蟺}{2}} (f (x) - g (x)) d x = {鈭�}_{- \frac{蟺}{2}}^{\frac{蟺}{4}} (\cos x - \sin x) d x + {鈭�}_{\frac{蟺}{4}}^{\frac{蟺}{2}} (\sin x - \cos x) d x = [\sin x + \cos x]_{- \frac{蟺}{2}}^{\frac{蟺}{4}} + [- \cos x - \sin x]_{\frac{蟺}{4}}^{\frac{蟺}{2}} = [(\sin \frac{蟺}{4} + \cos \frac{蟺}{4}) - (\sin (- \frac{蟺}{2}) + \cos (- \frac{蟺}{2}))] - [(\cos \frac{蟺}{4} + \sin \frac{蟺}{4}) - (\cos \frac{蟺}{2} + \sin \frac{蟺}{2})] = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} - (- 1 + 0) - \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} + 0 + 1 = 2$

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Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Graph of the functions.

Step 3. Required area.

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91影视

For each pair of functions fand gand interval [a,b]in Exercises 41-52, use definite integrals and the Fundamental Theorem of Calculus to find the exact area of the region between the graphs of fand gfrom x=atox=b. f(x)=sinx,g(x)=cosx,[a,b]=-蟺2,蟺2

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Graph of the functions.

Step 3. Required area.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Applied Mathematics

Pure Maths

Geometry

Mechanics Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.