1) Introduction: Integrals in Action 📚
Integrals aren’t just about areas—they help us solve real problems! In this topic, we’ll use definite integrals to calculate accumulated change (like total distance or water flow) and the average value of a function over an interval. These ideas are key in physics, economics, and more!
We’ll explore:
- Accumulated Change: Total change based on a rate.
- Average Value: The mean value of a function.
- Examples: Practical applications step-by-step.
Quick Recap: \( \int_a^b f(x) \, dx = F(b) - F(a) \) where \( F'(x) = f(x) \).
2) Accumulated Change 🎓
If \( f(x) \) is a rate of change (e.g., velocity), the definite integral \( \int_a^b f(x) \, dx \) gives the total accumulated change (e.g., distance) over \([a, b]\).
Definition 13.1: Accumulated Change
The accumulated change from \( a \) to \( b \) is \( \int_a^b f(x) \, dx \), where \( f(x) \) is the rate of change.
Example 1: Distance Traveled
A car’s velocity is \( v(t) = 2t \) m/s. Find the distance traveled from \( t = 0 \) to \( t = 3 \).
- Integrate: \( \int_0^3 2t \, dt \).
- Antiderivative: \( t^2 \).
- Evaluate: \( \left[ t^2 \right]_0^3 = 3^2 - 0^2 = 9 \).
Answer: Distance = 9 meters.
Example 2: Water Flow
The rate of water flow is \( f(t) = 3t^2 \) liters/min. Find total flow from \( t = 1 \) to \( t = 2 \).
- Integrate: \( \int_1^2 3t^2 \, dt \).
- Antiderivative: \( t^3 \).
- Evaluate: \( \left[ t^3 \right]_1^2 = 2^3 - 1^3 = 8 - 1 = 7 \).
Answer: 7 liters.
3) Average Value of a Function 📐
The average value of a function \( f(x) \) over \([a, b]\) is the total area under the curve (divided by the interval length) using the integral.
Definition 13.2: Average Value
The average value of \( f(x) \) on \([a, b]\) is \( \frac{1}{b - a} \int_a^b f(x) \, dx \).
Example 3: Average Velocity
Find the average velocity of \( v(t) = t^2 \) from \( t = 0 \) to \( t = 2 \).
- Integrate: \( \int_0^2 t^2 \, dt \).
- Antiderivative: \( \frac{t^3}{3} \).
- Evaluate: \( \left[ \frac{t^3}{3} \right]_0^2 = \frac{8}{3} \).
- Average: \( \frac{1}{2 - 0} \cdot \frac{8}{3} = \frac{8}{6} = \frac{4}{3} \).
Answer: Average velocity = \( \frac{4}{3} \) m/s.
Example 4: Average Temperature
The temperature \( T(t) = t + 10 \) °C over \( t = 0 \) to \( t = 5 \). Find the average.
- Integrate: \( \int_0^5 (t + 10) \, dt \).
- Antiderivative: \( \frac{t^2}{2} + 10t \).
- Evaluate: \( \left[ \frac{t^2}{2} + 10t \right]_0^5 = (12.5 + 50) - 0 = 62.5 \).
- Average: \( \frac{1}{5 - 0} \cdot 62.5 = 12.5 \).
Answer: Average temperature = 12.5 °C.
4) Advanced Examples 🔍
Example 5: Accumulated Change with Negatives
A particle’s velocity is \( v(t) = t - 2 \) m/s. Find displacement from \( t = 0 \) to \( t = 4 \).
- Integrate: \( \int_0^4 (t - 2) \, dt \).
- Antiderivative: \( \frac{t^2}{2} - 2t \).
- Evaluate: \( \left[ \frac{t^2}{2} - 2t \right]_0^4 = (8 - 8) - 0 = 0 \).
Answer: Displacement = 0 meters (net change).
Example 6: Average with Trigonometric Function
Find the average value of \( f(x) = \sin(x) \) from 0 to \( \pi \).
- Integrate: \( \int_0^\pi \sin(x) \, dx \).
- Antiderivative: \( -\cos(x) \).
- Evaluate: \( \left[ -\cos(x) \right]_0^\pi = -\cos(\pi) - (-\cos(0)) = 1 - (-1) = 2 \).
- Average: \( \frac{1}{\pi - 0} \cdot 2 = \frac{2}{\pi} \).
Answer: Average value = \( \frac{2}{\pi} \).
5) Practice Questions 🎯
Fundamental Practice Questions 🌱
Instructions: Compute the accumulated change or average value using definite integrals. 📚
\( \int_0^2 3t \, dt \) (velocity in m/s)
\( \int_1^4 x^2 \, dx \) (rate of water flow in L/min)
\( \int_0^3 (2x + 1) \, dx \) (speed in km/h)
\( \int_{-1}^2 (x - 1) \, dx \) (velocity in m/s)
\( \frac{1}{2} \int_0^2 x \, dx \) (average value)
\( \int_0^1 4t^2 \, dt \) (rate of change in units/min)
\( \frac{1}{3} \int_0^3 \cos(t) \, dt \) (average temperature in °C)
\( \int_{-2}^2 t^3 \, dt \) (velocity in m/s)
\( \int_0^4 \sqrt{t} \, dt \) (growth rate in cm/day)
\( \frac{1}{\pi} \int_0^\pi \sin(t) \, dt \) (average value)
\( \int_1^2 (t^2 + 2t) \, dt \) (rate of production in units/h)
Challenging Practice Questions 🌟
Instructions: These involve complex scenarios or interpretations. 🧠
Find the total distance traveled if velocity \( v(t) = t^2 - 4t + 3 \) from \( t = 0 \) to \( t = 5 \).
Compute the average value of \( f(x) = x^3 - x \) from -1 to 1.
Determine the accumulated change for \( r(t) = \sin(t) \) from 0 to \( 2\pi \).
Evaluate the average power output if \( P(t) = 2t e^{-t} \) from \( t = 0 \) to \( t = 2 \).
Find the total rainfall if the rate is \( R(t) = e^{-t} \) mm/h from \( t = 0 \) to \( t = 3 \).
6) Summary & Cheat Sheet 📋
6.1) Accumulated Change
\( \int_a^b f(x) \, dx \) gives total change if \( f(x) \) is a rate.
6.2) Average Value
\( \frac{1}{b - a} \int_a^b f(x) \, dx \) is the average over \([a, b]\).
6.3) Tip
Use absolute values for total distance if velocity changes sign.
You’ve applied integrals to real life! Next, we’ll calculate volumes. 🎉