A function that represents the accumulated amount of a quantity as its rate of change is integrated over time.

The area under the curve of a function between two specified limits.

A function that describes how a quantity changes with respect to another variable, often time.

The net change in position of an object.

Displacement (or net change in position).

The value of a function at a specific point, often at time t=0, used to determine the constant of integration.

The difference between the final and initial values of a quantity.

Units of the dependent variable per unit of the independent variable (e.g., mosquitoes per day).

Velocity is the derivative of position, and acceleration is the derivative of velocity.

To find the net change or accumulation of a quantity over a specific interval.

$\int_{a}^{b} v(t) dt$

$A(t) = A(0) + \int_{0}^{t} R(x) dx$

$\int_{a}^{b} f(x) dx$

$\int x^n dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$

$F(b) - F(a) = \int_{a}^{b} F'(x) dx$

$\int \cos(x) dx = \sin(x) + C$

$\int \sin(x) dx = -\cos(x) + C$

$1000 + \int_{0}^{31} R(t) dt$

$\int k dx = kx + C$

$\int kf(x) dx = k \int f(x) dx$

1. Identify the velocity function v(t). 2. Set up the definite integral: $\int_{a}^{b} v(t) dt$. 3. Evaluate the integral to find the displacement.

1. Identify the rate of change function R(t) and the initial condition A(0). 2. Set up the integral: $A(t) = A(0) + \int_{0}^{t} R(x) dx$. 3. Evaluate the integral and add it to the initial condition.

1. Identify the relevant interval. 2. Approximate the area under the curve using geometric shapes or numerical methods (e.g., trapezoidal rule). 3. Add the area to any given initial condition.

1. Identify the rate of change function R(t). 2. Set up the definite integral to find the net change: $\int_{0}^{31} R(t) dt$. 3. Add the result to the initial number of mosquitoes (1000). 4. Round to the nearest whole number.

1. Identify the rate of change function. 2. Set up the definite integral over the interval. 3. Evaluate the integral, considering the sign of the function to determine net change.

1. Use a numerical integration method (Trapezoidal Rule, Riemann Sum) to approximate the definite integral using the table values. 2. Add the result to any given initial condition.

1. Set up the accumulation function with the unknown time as the upper limit of integration. 2. Set the accumulation function equal to the target value. 3. Solve for the unknown time using algebraic methods or a calculator.

1. Find the critical points of the rate of change function by setting its derivative equal to zero. 2. Evaluate the accumulation function at the critical points and endpoints of the interval. 3. Compare the values to determine the maximum or minimum accumulated amount.

1. Break the integral into separate integrals corresponding to each piece of the function. 2. Evaluate each integral over its respective interval. 3. Sum the results to find the total accumulation.

1. Set up the definite integral of the rate of change function over the interval. 2. Divide the result by the length of the interval to find the average value.