Math Problem Statement
Solution
To find the average value of the function over the interval , we use the formula for the average value of a function over an interval :
Here, , , and . Let's compute this step by step.
Step 1: Find the integral of
We need to compute the indefinite integral of :
Step 2: Evaluate the definite integral
Now we evaluate the definite integral from 2 to 5:
First, calculate at :
Next, calculate at :
Now subtract:
Step 3: Compute the average value
Now divide by the length of the interval :
Thus, the average value of the function over the interval is:
Would you like any further details or have any other questions?
Here are five related questions to explore:
- How does the average value formula apply to functions that are negative on certain intervals?
- What is the geometric interpretation of the average value of a function over an interval?
- Can you find the average value of over a different interval, say ?
- How does the presence of a cubic term in the function affect the shape of the graph?
- What role does definite integration play in finding the area under curves?
Tip: When working with definite integrals, always carefully evaluate the function at the upper and lower limits to avoid errors.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Calculus
Integration
Average value of a function
Formulas
Average Value = (1 / (b - a)) * ∫[a to b] f(x) dx
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
Grades 11-12