Glossary

If a baker starts with 0 dough and adds flour at a rate of 100g per minute, the accumulation function would tell you the total amount of dough at any given time 'x'.

If $A(x) = \int_{0}^{x} R(t) , dt$ represents the total amount of water accumulated in a tank from time 0 to $x$, then $A(x)$ is an accumulation function.

If a car's speed (rate of change of distance) is 60 km/h for 2 hours, the accumulation of change in distance is 120 km.

If a company's profit rate is $P'(t)$ dollars per day, then $\int_{0}^{30} P'(t) , dt$ represents the total accumulation of change in profit over a month.

If a reservoir's water level changes by a certain rate each day, the accumulation of change over a week would be the total rise or fall in water level during that week.

If you know the rate at which water is flowing into a tank, the accumulation of change would tell you the total volume of water added over a specific time period.

If you need to find the total area under a curve from $x=0$ to $x=10$, you can use the adjacent intervals property to calculate it as $\int_{0}^{5} f(x) , dx + \int_{5}^{10} f(x) , dx$.

If $f(x) = 3x^2$, then $F(x) = x^3 + C$ is an antiderivative of $f(x)$, where C is any constant.

If the velocity of a particle is $v(t) = 2t$, then its position function $s(t) = t^2 + C$ is an Antiderivative, as differentiating $s(t)$ gives $v(t)$.

Finding the area under the curve of a power output function over time would give the total energy consumed.

If you integrate $f(x) = x^2$ from $x=0$ to $x=2$, the result is the area under the curve of the parabola from the origin to $x=2$.

If a graph shows the flow rate of water into a pool, calculating the area under the rate of change curve from t=0 to t=30 minutes tells you the total volume of water added in that half-hour.

If a bank account starts with $1000, that's the boundary value before any interest accrues or withdrawals are made.

A function like $f(x) = -x^2$ is concave down, indicating that a trapezoidal sum will underestimate the area under its curve.

For a function like $f(x) = x^2$, it is concave up over its entire domain, which means a trapezoidal sum will overestimate its area.

The integral $\int_{0}^{1} 5x^3 , dx$ can be simplified to $5 \int_{0}^{1} x^3 , dx$ using the constant multiple property, making calculations easier.

When finding the antiderivative of $2x$, we get $x^2 + C$; the Constant of Integration 'C' signifies that any constant term would vanish upon differentiation.

The function $f(x) = x^2$ is a continuous function over all real numbers, unlike a step function which has jumps.

The function $f(x) = -x^2$ for $x > 0$ is a decreasing function, as its graph goes down from left to right in that region.

To find the total distance traveled by a rocket given its velocity function from launch to 10 seconds, you would use a definite integral of the velocity function from t=0 to t=10.

Calculating the definite integral of a velocity function over a time interval gives the exact displacement of an object.

If a car's velocity is given by $v(t) = 2t$ m/s, the definite integral $\int_{0}^{5} 2t , dt$ would tell you the total distance traveled by the car in the first 5 seconds.

To find the total work done by a variable force from point A to point B, you would use a definite integral of the force function over that distance.

Calculating the total displacement of a car from time $t=0$ to $t=10$ seconds, given its velocity function, involves computing a definite integral.

The definite integral of a velocity function over a time interval gives the exact displacement of an object during that time.

Calculating the total distance traveled by a car with varying speed over a specific time period involves evaluating a Definite Integral of its velocity function.

In $\int_{0}^{x} \sin(t) dt$, 't' is the dummy variable used to perform the integration, while 'x' remains the variable of the resulting function.

To find the exact area under the curve of $f(x) = 2x$ from $x=1$ to $x=3$, you'd use the First Fundamental Theorem of Calculus by finding its antiderivative $F(x) = x^2$ and calculating $F(3) - F(1)$.

In the integral $\int_{2}^{x} f(t) dt$, the number 2 is the fixed starting point (a).

Understanding the Fundamental Theorem of Calculus allows you to solve problems that involve both rates of change and total accumulation, like finding the total distance traveled from a velocity function.

Using the Fundamental Theorem of Calculus, you can quickly find the exact area under the curve of $f(x) = x^2$ from 0 to 1 by evaluating its antiderivative at those points.

When calculating the total distance traveled by a car, the odometer reading at the beginning of the trip serves as the given starting point.

The heights of the trapezoids are crucial because they represent the function's magnitude at the specific x-values, directly influencing the area calculation.

The function $f(x) = x^3$ is an increasing function over all real numbers, meaning its graph always goes up from left to right.

Finding the Indefinite Integral of $f(x) = 3x^2$ gives $x^3 + C$, representing all functions whose derivative is $3x^2$.

In the integral ${\int }_{a}^{b} \sin(x) , dx$, $\sin(x)$ is the Integrand, the function whose area under the curve is being found.

In the integral $\int_{1}^{3} (x^2 + 2x) , dx$, the integrand is $x^2 + 2x$, which is the function whose area or accumulation we are finding.

For $\int_{0}^{1} e^y , dy$, the integration variable is $dy$, meaning we are integrating with respect to $y$, not $x$ or any other variable.

When calculating the work done by a variable force from $x=1$ meter to $x=5$ meters, the Interval of integration is $[1, 5]$.

When approximating the area under $f(x) = x^2$ from $x=0$ to $x=4$ using four rectangles, the first rectangle's height would be $f(0)$, demonstrating a Left Riemann Sum.

In $\int_{2}^{5} \sin(x) , dx$, the limits of integration are $a=2$ and $b=5$, meaning we are calculating the net area from $x=2$ to $x=5$.

In ${\int }_{0}^{5} x^2 , dx$, the value 0 is the Lower Limit, defining the leftmost boundary of the area being calculated.

When calculating the average temperature over a day, a Midpoint Riemann Sum would use the temperature reading from the middle of each hour to represent that hour's average.

If the temperature of a cooling object is dropping, it exhibits a negative rate of change in temperature.

If you use a trapezoidal sum to approximate the area under a concave up curve, your result will be an overestimate of the true area.

For a decreasing function, a Left Riemann Sum will typically result in an overestimate of the true area under the curve.

When a company's profit is growing, it has a positive rate of change in profit, meaning more money is being earned.

A factory produces 50 widgets per hour; this is its rate of change of production.

If $R(t)$ is the rate at which water flows into a tank in liters per minute, then $\int_{0}^{10} R(t) , dt$ gives the total volume of water accumulated in the tank over the first 10 minutes, illustrating the rate of change interpretation.

If a population grows at a certain number of individuals per year, that growth rate is the rate of change function for the population.

If $\int_{1}^{3} f(x) , dx = 7$, then $\int_{3}^{1} f(x) , dx = -7$ due to the reversing limits property.

To estimate the area under $f(x) = x^2$ from $0$ to $1$, one could use a Riemann sum with 4 rectangles, calculating the sum of their areas to get an approximation.

To estimate the total distance a car traveled given its varying speed over time, one could use a Riemann sum by adding up the areas of rectangles representing speed multiplied by small time intervals.

To approximate the total work done by a variable force, a Right Riemann Sum might use the force at the end of each small displacement interval to calculate the work for that segment.

If you have a function defined as $G(x) = \int_{5}^{x} \sin(t) , dt$, the Second Fundamental Theorem of Calculus immediately tells you that $G'(x) = \sin(x)$.

For $f(x) = \cos(x)$ from $0$ to $2\pi$, the integral $\int_{0}^{2\pi} \cos(x) , dx = 0$ because the positive area above the x-axis from $0$ to $\pi/2$ and $3\pi/2$ to $2\pi$ is cancelled out by the negative area below the x-axis from $\pi/2$ to $3\pi/2$, demonstrating the concept of signed area.

If you're integrating from $x=0$ to $x=10$ and divide the interval into 5 equal parts, each part like $[0,2]$ or $[2,4]$ is a subinterval.

If you're estimating the area under a curve from $x=0$ to $x=10$, you might divide it into five subintervals like $[0,2], [2,4], [4,6], [6,8], [8,10]$.

To get a more accurate approximation of an integral, you would typically divide the main interval into a greater number of subintervals.

To integrate $(x^2 + \sin(x))$ from $0$ to $\pi$, you can use the sum/difference property to evaluate $\int_{0}^{\pi} x^2 , dx + \int_{0}^{\pi} \sin(x) , dx$ separately.

To estimate the total distance a car traveled given its speed at various times, you could use a trapezoidal sum on the speed-time graph.

When approximating the area under a curve, each segment is treated as a trapezoid where the parallel sides are the function values at the interval endpoints.

When approximating the area under a concave down curve using a trapezoidal sum, the calculated value will be an underestimate of the actual area.

For an increasing function, a Left Riemann Sum will typically result in an underestimate of the true area under the curve.

If a population grows at a rate of 'people per year' over 'years', the units of accumulation will be 'people', representing the total change in population.

In ${\int }_{0}^{5} x^2 , dx$, the value 5 is the Upper Limit, defining the rightmost boundary of the area being calculated.

When calculating the volume of water flowing into a tank over time, 'x' represents the variable ending point (x), allowing you to find the volume at any future moment.

In a function describing the total rainfall over time, 'x' represents the variable endpoint, allowing you to find the total rainfall up to any specific day.

In a trapezoidal sum with equal width subintervals, the total interval length is simply divided by the number of trapezoids.

If you integrate $f(x) = x^2 + 3$ from $x=5$ to $x=5$, the result is $0$ because of the zero-length interval property, as there is no area over a single point.