Q 82. Consider the function f(x)=lnx.(... [FREE SOLUTION]

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Chapter 5: Q 82. (page 431)

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

Short Answer

Expert verified

Part (a). The solution is $- \frac{7}{2} + \frac{1}{2} \ln (2) + 4 \ln (4)$ .

Part (b). The solution is $- 7 + \frac{1}{2} \ln (2) + 4 \ln (4)$ .

Step by step solution

Part (a) Step 1. Given information.

The given function is $f (x) = \ln x$ .

Part (a) Step 2. Solve the definite integral ∫124lnx dx.

Apply integration by parts and solve.

$\begin{array}{rcl} {鈭�}_{\frac{1}{2}}^{4} \ln x d x & = & {[\ln x 鈭� d x - 鈭� \{\frac{d}{d x} (\ln x) 鈭� d x\} d x]}_{\frac{1}{2}}^{4} \\ = & {[\ln x 路 x - 鈭� \frac{1}{x} 路 x d x]}_{\frac{1}{2}}^{4} \\ = & {[\ln x 路 x - 鈭� d x]}_{\frac{1}{2}}^{4} \\ = & {[\ln x 路 x - x]}_{\frac{1}{2}}^{4} \\ = & [4 \ln 4 - 4] - [\frac{1}{2} \ln \frac{1}{2} - \frac{1}{2}] \\ = & - \frac{7}{2} + \frac{1}{2} \ln (2) + 4 \ln (4) \end{array}$

Part (b) Step 1. Given information.

The given function is $f (x) = \ln x$ and $g (x) = 1$ .

Part (b) Step 2. Find the area between the graph of f(x)=lnx and g(x)=1 on 12,4.

The area is calculated below.

$\begin{array}{rcl} {鈭�}_{\frac{1}{2}}^{4} (f (x) - g (x)) d x & = & {鈭�}_{\frac{1}{2}}^{4} [\ln x - 1] d x \\ = & {鈭�}_{\frac{1}{2}}^{4} \ln x d x - {鈭�}_{\frac{1}{2}}^{4} d x \\ = & {[x \ln x - x]}_{\frac{1}{2}}^{4} - {[x]}_{\frac{1}{2}}^{4} \\ = & \frac{- 7}{2} + \frac{1}{2} \ln (2) + 4 \ln (4) - [4 - \frac{1}{2}] \\ = & \frac{- 7}{2} + \frac{1}{2} \ln (2) + 4 \ln (4) - \frac{7}{2} \\ = & - \{\frac{7 + 7}{2}\} + \frac{1}{2} \ln (2) + 4 \ln (4) \\ = & - 7 + \frac{1}{2} \ln (2) + 4 \ln (4) \end{array}$

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Short Answer

Step by step solution

Part (a) Step 1. Given information.

Part (a) Step 2. Solve the definite integral ∫124lnx dx.

Part (b) Step 1. Given information.

Part (b) Step 2. Find the area between the graph of f(x)=lnx and g(x)=1 on 12,4.

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Consider the function f(x)=lnx.(a) Find the signed area between the graph of f(x)and the x-axis on 12,4shown in the figure next at the left.(b) Find the area between the graph of f(x)and the graph of g(x)=1on 12,4shown in the figure next at the right.

Short Answer

Step by step solution

Part (a) Step 1. Given information.

Part (a) Step 2. Solve the definite integral ∫124lnx dx.

Part (b) Step 1. Given information.

Part (b) Step 2. Find the area between the graph of f(x)=lnx and g(x)=1 on 12,4.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Calculus

Pure Maths

Theoretical and Mathematical Physics

Geometry

Discrete Mathematics

Study anywhere. Anytime. Across all devices.