What is Displacement as an Integral?

S7-SA1-0142

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

Displacement as an integral is how we find the total change in position of an object when its speed (velocity) is changing over time. It's like adding up tiny steps an object takes, where each step's length depends on its speed at that exact moment. We use integration to sum up these tiny changes to get the total displacement.

Simple Example

Quick Example

Imagine an auto-rickshaw driver in Delhi who speeds up and slows down during a 10-minute trip. If you know the auto's speed at every single second, you can use integration to calculate the exact total distance (displacement) it covered from the starting point to the end point. It's not just speed times time because the speed isn't constant.

Worked Example

Step-by-Step

Let's say a cricket ball's velocity (speed with direction) is given by the function v(t) = 3t^2 - 2t meters per second, where 't' is time in seconds. We want to find the displacement of the ball between t = 1 second and t = 3 seconds.

Step 1: Understand that displacement is the integral of velocity with respect to time. So, Displacement = ∫v(t) dt.
---Step 2: Set up the definite integral with the given time limits: Displacement = ∫[from 1 to 3] (3t^2 - 2t) dt.
---Step 3: Find the antiderivative (the integral) of 3t^2 - 2t. The integral of 3t^2 is 3 * (t^3 / 3) = t^3. The integral of -2t is -2 * (t^2 / 2) = -t^2.
---Step 4: So, the antiderivative is F(t) = t^3 - t^2.
---Step 5: Now, evaluate F(t) at the upper limit (t=3) and the lower limit (t=1), and subtract: Displacement = F(3) - F(1).
---Step 6: Calculate F(3) = (3)^3 - (3)^2 = 27 - 9 = 18.
---Step 7: Calculate F(1) = (1)^3 - (1)^2 = 1 - 1 = 0.
---Step 8: Subtract: Displacement = 18 - 0 = 18 meters.

Answer: The displacement of the cricket ball between t=1s and t=3s is 18 meters.

Why It Matters

This concept is crucial for understanding how things move and change. Engineers use it to design safe cars and rockets, while AI/ML specialists use similar ideas to track movement patterns. It's also vital in climate science to model how pollutants spread or how glaciers melt over time.

Common Mistakes

MISTAKE: Confusing displacement with total distance. | CORRECTION: Displacement is the net change in position (can be negative if you end up behind your start), while total distance is the sum of all path lengths, always positive. Displacement is ∫v(t) dt, total distance is ∫|v(t)| dt.

MISTAKE: Forgetting to include the constant of integration when finding the indefinite integral. | CORRECTION: When calculating definite integrals for displacement, the constant 'C' cancels out, so it's not needed. However, remember it for indefinite integrals.

MISTAKE: Incorrectly applying the limits of integration (upper minus lower). | CORRECTION: Always substitute the upper limit first into the antiderivative, then subtract the result of substituting the lower limit. F(b) - F(a).

Practice Questions

Try It Yourself

QUESTION: If the velocity of a small drone is given by v(t) = 4t meters per second, find its displacement from t=0 to t=2 seconds. | ANSWER: 8 meters

QUESTION: A bullet's velocity is v(t) = 6t^2 - 10t + 4 meters per second. What is its displacement between t=0 and t=1 second? | ANSWER: 3 meters

QUESTION: The velocity of a train approaching a station is v(t) = 12 - 2t meters per second. Find the total distance covered (not just displacement) by the train from t=0 to t=8 seconds. (Hint: find when v(t) changes sign). | ANSWER: 40 meters

MCQ

Quick Quiz

If the velocity of a particle is v(t) = 2t + 1, what is its displacement from t=0 to t=2?

2 units

4 units

6 units

8 units

The Correct Answer Is:

To find displacement, integrate v(t) = 2t + 1 from 0 to 2. The integral is t^2 + t. Evaluating at the limits: (2^2 + 2) - (0^2 + 0) = (4 + 2) - 0 = 6.

Real World Connection

In the Real World

Delivery services like Swiggy or Zomato use complex algorithms that involve integration concepts to estimate delivery times and optimize routes. By knowing how the delivery rider's speed changes in traffic, they can calculate the displacement (distance covered) and predict arrival times more accurately, ensuring your food reaches you hot!

Key Vocabulary

Key Terms

INTEGRATION: A mathematical operation used to find the total sum of many small parts, like finding the area under a curve | VELOCITY: The speed of an object in a given direction | DISPLACEMENT: The overall change in position from a starting point to an ending point | DEFINITE INTEGRAL: An integral with upper and lower limits, used to find the exact value of an accumulation over an interval | ANTIDERIVATIVE: The reverse process of differentiation; the function whose derivative is the original function.

What's Next

What to Learn Next

Next, you can explore 'Applications of Integrals in Area and Volume'. This will show you how integration isn't just for movement, but also for calculating areas of complex shapes and volumes of 3D objects, which is super useful in engineering and design.