Accumulation of change refers to the total amount that a quantity has increased or decreased over a given interval
If \( f(x) \) represents the velocity of an object, then \( \int_a^b f(x) \, dx \) calculates the total displacement
True
To calculate the total distance traveled by a vehicle with velocity \( v(t) = 2t^2 \) m/s over the interval \( [1, 5] \) seconds, the definite integral is \( \int_1^5 2t^2 \, dt
The accumulation of change over an interval is mathematically calculated using a definite integral.
The definite integral of velocity calculates total displacement.
True
Steps to set up a definite integral to model the accumulation of change
1️⃣ Identify the rate function
2️⃣ Define the interval
3️⃣ Assemble the integral
What definite integral calculates the total distance traveled by a vehicle with velocity \( v(t) = 2t^2 \) m/s over the interval [1, 5] seconds?
\int_1^5 2t^2 \, dt
Steps to calculate a definite integral using the Fundamental Theorem of Calculus
1️⃣ Find the antiderivative
2️⃣ Evaluate at limits
3️⃣ Subtract values
Match the type of accumulation with its calculation and explanation:
Net Accumulation ↔️ \int_a^b f(x) \, dx, change considering direction
Total Accumulation ↔️ \int_a^b |f(x)| \, dx, overall change without direction
The accumulation of change is mathematically represented by an integral
True
The accumulation of a quantity over an interval can be calculated using a definite integral
Steps to set up a definite integral for modeling the accumulation of change
1️⃣ Identify the rate function
2️⃣ Define the interval
3️⃣ Assemble the integral
The definite integral \( \int_a^b f(t) \, dt \) calculates the total displacement if \( f(t) \) represents velocity
True
What does the accumulation of change measure over an interval?
Total increase or decrease
The start and end points of the interval in a definite integral are called the limits of integration.
Match the component of a definite integral with its description:
Limits of Integration ↔️ Start and end points of the interval
Integrand ↔️ Rate function defining the rate of change
Differential ↔️ Variable with respect to integration
The Fundamental Theorem of Calculus is used to calculate definite integrals by finding the antiderivative of the integrand.
What is the value of \( \int_1^3 (3x^2 + 2) \, dx \) using the Fundamental Theorem of Calculus?
30
To find the net amount of water added to a tank from \( t = 0 \) to \( t = 4 \) minutes with a rate of \( r(t) = 2t \) gallons per minute, the definite integral is \( \int_0^4 2t \, dt \), which equals 16 gallons.