Q. 57 Consider the function f(x)=1x2-4... [FREE SOLUTION]

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Chapter 5: Q. 57 (page 442)

Consider the function $f (x) = \frac{1}{x^{2} - 4}$
(a) Find the signed area between the graph of f(x) and thex-axis on [鈭�1, 1]shown next at the left.
(b) Find the area between the graph of f(x) and the graph of $g (x) = \frac{1}{4} (x - 1)$ on[鈭�1, 1] shown next at the right.

Short Answer

Expert verified

(a) The signed area between the graph of f(x)and x-axis is $\frac{\ln 3}{2}$ .

(b) Area between the graph of $f (x) a n d g (x)$ is $\frac{1}{4}$ .

Step by step solution

Part (a) Step 1. Given Information.

The function:

$f (x) = \frac{1}{x^{2} - 4}$

Part (b) Step 2. Write in terms of partial fraction.

$\frac{1}{x^{2} - 4} = \frac{1}{(x - 2) (x + 2)} = \frac{A}{(x - 2)} + \frac{B}{(x + 2)} = \frac{A (x + 2) + B (x - 2)}{x^{2} - 4} 1 = A (x + 2) + B (x - 2)$

Part (a) Step 3. Find A, B.

From the above equation,

$A + B = 1 2 A - 2 B = 1$

From the above equation,

$A = \frac{1}{4}, B = - \frac{1}{4}$

So the function is,

$\frac{1}{x^{2} - 4} = \frac{\frac{1}{4}}{x - 2} - \frac{\frac{1}{4}}{x + 2}$

Part (a) Step 4. Find the area.

$鈭� \frac{1}{x^{2} - 4} d x = 鈭� \frac{1}{4 (x - 2)} d x + 鈭� \frac{1}{4 (x + 2)} d x = \frac{1}{4} 鈭� \frac{1}{u} d u + \frac{1}{4} 鈭� \frac{1}{u} d u (w h e r e u = x 卤 2, d u = d x) = \frac{1}{4} \ln (|x - 2|) + \frac{1}{4} \ln (x + 2) + C$

Part (a)( Step 5. Substitute the limits.

Substitute the limits to get the area,

${鈭�}_{- 1}^{1} \frac{1}{x^{2} - 4} d x = {[\frac{1}{4} \ln (|x - 2|) + \frac{1}{4} \ln (x + 2) + C]}_{- 1}^{1} = \frac{\ln 3}{2}$

Part (b) Step 1. Find the area between f(x) and g(x).

The area of the region is:

$A = {鈭�}_{- 1}^{1} [\frac{1}{x^{2} - 4} - \frac{1}{4} (x - 1)] d x = {鈭�}_{- 1}^{1} \frac{1}{x^{2} - 4} d x - \frac{1}{4} {鈭�}_{- 1}^{1} x d x + \frac{1}{4} {鈭�}_{- 1}^{1} d x = [\frac{1}{4} \ln (|x - 2|) + \frac{1}{4} \ln (x + 2) - \frac{1}{4} {[\frac{x^{2}}{2} - x]]}_{- 1}^{1} = \frac{1}{4}$

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91影视

Consider the function $f (x) = \frac{1}{x^{2} - 4}$
(a) Find the signed area between the graph of f(x) and thex-axis on [鈭�1, 1]shown next at the left.
(b) Find the area between the graph of f(x) and the graph of $g (x) = \frac{1}{4} (x - 1)$ on[鈭�1, 1] shown next at the right.

Short Answer

Step by step solution

Part (a) Step 1. Given Information.

Part (b) Step 2. Write in terms of partial fraction.

Part (a) Step 3. Find A, B.

Part (a) Step 4. Find the area.

Part (a)( Step 5. Substitute the limits.

Part (b) Step 1. Find the area between f(x) and g(x).

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Consider the function f(x)=1x2-4(a) Find the signed area between the graph of f(x) and thex-axis on [鈭�1, 1]shown next at the left.(b) Find the area between the graph of f(x) and the graph of g(x)=14(x-1) on[鈭�1, 1] shown next at the right.

Short Answer

Step by step solution

Part (a) Step 1. Given Information.

Part (b) Step 2. Write in terms of partial fraction.

Part (a) Step 3. Find A, B.

Part (a) Step 4. Find the area.

Part (a)( Step 5. Substitute the limits.

Part (b) Step 1. Find the area between f(x) and g(x).

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Logic and Functions

Probability and Statistics

Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

Consider the function $f (x) = \frac{1}{x^{2} - 4}$
(a) Find the signed area between the graph of f(x) and thex-axis on [鈭�1, 1]shown next at the left.
(b) Find the area between the graph of f(x) and the graph of $g (x) = \frac{1}{4} (x - 1)$ on[鈭�1, 1] shown next at the right.