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.

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

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.

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

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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

Applied Mathematics

Theoretical and Mathematical Physics

Pure Maths

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Calculus

Mechanics Maths

Study anywhere. Anytime. Across all devices.