Definite integrals. Barrow's rule
So far you have been finding antiderivatives: families of functions with a loose constant $+C$. A definite integral is something different: it is a specific number, the result of evaluating an antiderivative between two limits. Among other things, that number measures how much a quantity (income, population, cost) has grown between two instants.
The definite integral and Barrow's rule
Calculate the value of the following definite integral by applying Barrow's rule, which represents the area under the parabola $f(x) = x^2 + 2$ in the interval $[0, 3]$:…
Calculate the value of the following definite integral by applying Barrow's rule, which represents the area under the parabola $f(x) = x^2 + 2$ in the interval $[0, 3]$:
$$\int_{0}^{3} (x^2 + 2) \, dx$$Find the value of the definite integral of the basic exponential function between the values 0 and 1: $$\int_{0}^{1} e^x \, dx$$ Note: Express the result in terms of the…
Find the value of the definite integral of the basic exponential function between the values 0 and 1:
$$\int_{0}^{1} e^x \, dx$$Note: Express the result in terms of the number $e$.
A courier company has found that the marginal cost of processing its parcels (in euros per parcel) is given by the function $C'(x) = 0,06x + 2$, where $x$ is the number…
A courier company has found that the marginal cost of processing its parcels (in euros per parcel) is given by the function $C'(x) = 0,06x + 2$, where $x$ is the number of parcels. Calculate the increase in total cost if the number of parcels processed increases from 10 to 20 using the integral:
$$\int_{10}^{20} (0,06x + 2) \, dx$$Properties: additivity and piecewise functions
When you know the value of some integrals, you can often work out others without calculating any antiderivative.
Given a continuous function $f(x)$, the following values of its definite integrals are known: $\int_{1}^{5} f(x) \, dx = 12$ $\int_{1}^{3} f(x) \, dx = 4$ Using the…
Given a continuous function $f(x)$, the following values of its definite integrals are known:
-
$\int_{1}^{5} f(x) \, dx = 12$
-
$\int_{1}^{3} f(x) \, dx = 4$
Using the properties of the definite integral, calculate the value of $\int_{3}^{5} f(x) \, dx$.
Consider the piecewise-defined function: $f(x) = \begin{cases} 2x + 4 & \text{if } x < 1 \\ x^2 + 5 & \text{if } x \ge 1 \end{cases}$ Calculate the value of the definite…
Consider the piecewise-defined function:
$f(x) = \begin{cases} 2x + 4 & \text{if } x < 1 \\ x^2 + 5 & \text{if } x \ge 1 \end{cases}$
Calculate the value of the definite integral $\int_{0}^{2} f(x) \, dx$ by applying the additivity property of the integral with respect to the interval of integration.
The integral as an accumulated quantity
If a quantity changes at a certain rate (marginal cost, velocity, flow of customers), the total accumulated between two instants is the definite integral of that rate.
The flow of customers entering a shopping centre on a Saturday morning can be modelled by the function $f(t) = 100 + 20t$ (customers per hour), where $t$ is the time in…
The flow of customers entering a shopping centre on a Saturday morning can be modelled by the function $f(t) = 100 + 20t$ (customers per hour), where $t$ is the time in hours since opening ($t=0$). Calculate the total number of customers who entered between the second and fourth hours after opening:
$$\int_{2}^{4} (100 + 20t) \, dt$$A courier company has estimated that its rate of change of daily revenue (in hundreds of euros) is given by the function $I'(t) = 0,3t^2 - 2t + 10$, where $t$ is the…
A courier company has estimated that its rate of change of daily revenue (in hundreds of euros) is given by the function $I'(t) = 0,3t^2 - 2t + 10$, where $t$ is the time in hours since opening ($0 \le t \le 8$). Use Barrow's rule to calculate the total increase in revenue between the third hour ($t=3$) and the sixth hour ($t=6$) of its working day.
The fundamental theorem: functions defined by an integral
Sometimes the upper limit of an integral is not a fixed number, but a variable (or a function of one). The result is then a function, and the fundamental theorem of calculus tells you how to differentiate it without calculating any antiderivative.
Determine the value of the positive parameter $k$ so that the area enclosed by the function $f(x) = kx^2$ and the $OX$-axis between the lines $x = 0$ and $x = 3$ is…
Determine the value of the positive parameter $k$ so that the area enclosed by the function $f(x) = kx^2$ and the $OX$-axis between the lines $x = 0$ and $x = 3$ is exactly $18$ square units.
Let $f(x) = ax^2 + bx + c$ be a quadratic polynomial function. It is known to have a local extremum at the point $P(1, -4)$ and that the definite integral of the…
Let $f(x) = ax^2 + bx + c$ be a quadratic polynomial function. It is known to have a local extremum at the point $P(1, -4)$ and that the definite integral of the function over the interval $[0, 3]$ is equal to $0$.
- a) Determine the values of the parameters $a$, $b$ and $c$.
- b) Calculate the area of the region bounded by the graph of the function $f(x)$ and the x-axis $OX$.
Given the integral function $F(x) = \int_{1}^{x^2} \ln(t) \, dt$, defined for $x \geq 1$: a) Use the Fundamental Theorem of Calculus and the Chain Rule to find the…
Given the integral function $F(x) = \int_{1}^{x^2} \ln(t) \, dt$, defined for $x \geq 1$:
- a) Use the Fundamental Theorem of Calculus and the Chain Rule to find the derivative function $F'(x)$.
- b) Without calculating the integral explicitly, determine whether the function $F(x)$ is increasing or decreasing on its domain and justify whether it has any point of inflection.
- c) Calculate the exact value of $F(e)$ by integrating by parts.