Questions: What is the area beneath the curve (y=frac5-x^2x^3) from (x=1) to (x=2) ? (frac158-ln 2) (frac13) (frac358) (-frac152+ln 1)

$What is the area beneath the curve (y=frac5-x^2x^3) from (x=1) to (x=2) ? (frac158-ln 2) (frac13) (frac358) (-frac152+ln 1)$

Transcript text: What is the area beneath the curve $y=\frac{5-x^{2}}{x^{3}}$ from $x=1$ to $x=2$ ? $\frac{15}{8}-\ln 2$ $\frac{1}{3}$ $\frac{35}{8}$ $-\frac{15}{2}+\ln 1$

Solution

Solution Steps

To find the area beneath the curve $ y = \frac{5-x^2}{x^3} $ from $ x = 1 $ to $ x = 2 $, we need to compute the definite integral of the function over this interval. This involves integrating the function with respect to $ x $ and evaluating it from 1 to 2.

Step 1: Define the Function

The function given is $ y = \frac{5 - x^2}{x^3} $.

Step 2: Set Up the Definite Integral

To find the area beneath the curve from $ x = 1 $ to $ x = 2 $, we need to evaluate the definite integral:

\[ \int_{1}^{2} \frac{5 - x^2}{x^3} \, dx \]

Step 3: Evaluate the Integral

The integral of the function is calculated as:

\[ \int \frac{5 - x^2}{x^3} \, dx = \int \left( \frac{5}{x^3} - \frac{x^2}{x^3} \right) \, dx = \int \left( \frac{5}{x^3} - \frac{1}{x} \right) \, dx \]

This simplifies to:

\[ \int \left( 5x^{-3} - x^{-1} \right) \, dx \]

Step 4: Compute the Definite Integral

Evaluating the integral from $ x = 1 $ to $ x = 2 $:

\[ \left[ -\frac{5}{2}x^{-2} - \ln|x| \right]_{1}^{2} \]

Substituting the limits:

\[ \left( -\frac{5}{2} \cdot \frac{1}{4} - \ln 2 \right) - \left( -\frac{5}{2} \cdot 1 - \ln 1 \right) \]

Simplifying:

\[ \left( -\frac{5}{8} - \ln 2 \right) - \left( -\frac{5}{2} \right) \]

\[ = \frac{15}{8} - \ln 2 \]