Questions: The units are given for y and for f(x). Give the units of the integral from a to b of f(x) dx. x is a portion in "feet" and f(x) is an area in "square feet."

Transcript text: The units are given for $y$ and for $f(x)$. Give the units of $\int_{a}^{b} f(x) dx$ $x$ is a portion in "feet" and $f(x)$ is an area in "square feet."

Solution

Solution Steps

To determine the units of the integral $\int_{a}^{b} f(x) \, dx$, we need to understand the units of $f(x)$ and $dx$. The integral represents the area under the curve of $f(x)$ from $a$ to $b$. The units of $f(x)$ are given as "square feet" and the units of $x$ are given as "feet". Therefore, the units of the integral will be the product of these units.

Solution Approach

Identify the units of $f(x)$ and $dx$.
Multiply these units to get the units of the integral.

Step 1: Identify the Units of $ f(x) $ and $ dx $

The units of the function $ f(x) $ are given as "square feet", which can be expressed as $ \text{ft}^2 $. The differential $ dx $ represents a change in $ x $, which is measured in "feet" or $ \text{ft} $.

Step 2: Calculate the Units of the Integral

The integral $ \int_{a}^{b} f(x) \, dx $ represents the area under the curve of $ f(x) $ from $ a $ to $ b $. The units of the integral can be calculated by multiplying the units of $ f(x) $ and $ dx $:

\[ \text{Units of } \int_{a}^{b} f(x) \, dx = \text{Units of } f(x) \times \text{Units of } dx = \text{ft}^2 \times \text{ft} = \text{ft}^3 \]