Question
Given the function
for
The net area
The total area
for
The net area
The total area
Ask by Cross Savage. in the United States
Mar 21,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
Net area: 114
Total area: 118
Total area: 118
Solution
We are given
for
Step 1. Find the antiderivative
We integrate term‐by‐term:
Thus, an antiderivative
Step 2. Compute the Net Area
The net area is given by
First, compute
Next, compute
Thus, the net area is
Step 3. Determine Where the Function Changes Sign
We need the total area, which is the sum of the absolute areas where
Notice that
The factor
- For
, . - For
, .
Thus, the only sign change in the interval
Step 4. Compute the Total Area
The total area is
Since
Compute
Taking the absolute value gives
Compute
Therefore, the total area is
Final Answers
Net area:
Total area:
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Simplify this solution Mind Expander
To find the net area under the curve of the function
Calculating this integral gives:
Evaluating it at the endpoints yields:
Next, to find the total area, you need to consider the sections where the function falls below the x-axis. By checking for zeros of
You would find that
Thus, the total area will be the net area adding absolute values of any sections below the x-axis, traditionally:
Calculating those areas can be a bit involved, so be careful with your signs! Summing up relevant areas gives you the total area between
