Exercises: Definite integrals. 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 $[0, 3]$:

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:

Calculate the area of the plane region bounded by the line $f(x) = 2x - 4$, the $OX$-axis and the vertical lines $x = 2$ and $x = 5$. Take into account that in this interval the function is always positive or zero.

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 $\int_{3}^{5} f(x) \, dx$.

Find the value of the definite integral of the basic exponential function between the values 0 and 1:

Note: Express the result in terms of the number $e$.

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:

Calculate the area enclosed by the curve $f(x) = 4 - x^2$ and the x-axis ($OX$).

Calculate the value of the following definite integral of a simple rational function:

Remember that $\ln(e) = 1$ and $\ln(1) = 0$.

Javier and Lucía are calculating the definite integral $\int_1^3 (4x - 1)\,dx$ in class. Javier chooses as antiderivative $F(x) = 2x^2 - x$ and obtains a numerical result. Lucía, on the other hand, decides to use the antiderivative $G(x) = 2x^2 - x + 5$ and claims with conviction: "My result has to be 5 units greater than yours, because my antiderivative is worth exactly 5 units more at any point". a) Explain Lucía's conceptual error with reasoning, demonstrating, through the theoretical expression of Barrow's rule with any constant $+C$, why the constant always cancels out. b) Calculate the value of the definite integral by first evaluating Javier's antiderivative and then Lucía's, verifying that both approaches necessarily lead to the same numerical value.

Consider the function $f(x) = x^2 + 1$ on the interval $[1, 3]$. Since the function is continuous and strictly increasing on this interval, the area under its curve (given by the integral $\int_1^3 (x^2 + 1)\,dx$) must be bounded between the area of a smaller rectangle (with base equal to the length of the interval and height equal to the minimum value $f(1)$) and the area of a larger rectangle (with the same base and height equal to the maximum value $f(3)$). a) Calculate the areas of both rectangles to provide a preliminary estimate: between which two integer values must the result of the integral lie? b) Apply Barrow's rule to calculate the exact value of the definite integral. c) Verify that the exact value obtained satisfies the estimated bounds and calculate what percentage of the larger rectangle's area is represented by the exact value accumulated under the curve.

Calculate the area of the plane region bounded by the graph of the function $f(x) = x^2 - 4x + 3$, the x-axis $OX$ and the vertical lines $x = 0$ and $x = 2$. Take into account the position of the curve relative to the axis in the given interval to perform the calculation correctly.

Find the area of the region bounded by the graphs of the functions $f(x) = -x^2 + 4x$ and $g(x) = x$. To do so, first determine the points of intersection of both curves and sketch the region to identify which of the functions lies above the other in the interval of integration.

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.

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.

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.

A graphic designer is creating a new logo whose upper outline is defined by the function $f(x) = 4 - \frac{x^2}{4}$ and whose lower base is defined by the $OX$-axis. If the logo lies between $x = -2$ and $x = 2$, calculate the total surface area of the logo using the definite integral.

Find the area of the finite region bounded by the curve $y = \frac{1}{x}$, the x-axis and the lines $x = 1$ and $x = e^2$. Express the result in square units in exact form using logarithms.

Calculate the area of the region bounded by the graph of the function $f(x) = x^3 - x$ and the $OX$-axis. Note that the function changes sign in the natural interval defined by its roots, so you must identify these roots to set up the necessary integrals.

Due to a summer drought in a river basin in southern Spain, the rate of change of the volume of dammed water in a reservoir (measured in cubic hectometres per week, $\text{hm}^3\text{/week}$) over the 6 central weeks of the summer ($t \in [0, 6]$) follows the model: $$v(t) = 0,3t^2 - 1,8t$$ a) Justify analytically, by studying the sign of $v(t)$ on the interval $(0, 6)$ without calculating any integral, why the reservoir continuously loses water during those six weeks. b) Apply Barrow's rule to calculate $\int_0^6 v(t)\,dt$ and interpret both the algebraic sign and the numerical value obtained in relation to the dammed water. c) If at the beginning of the period ($t=0$) the reservoir had $45\text{ hm}^3$, determine the volume of water remaining at the end of the sixth week and find in which specific week the rate of loss was most severe.

The customer retention department of a telecommunications company in Spain analyses the rate of contract cancellations following a rate change. The rate at which cancellations occur (in hundreds of customers per month) is modelled by the function $f(t) = 6 - 0,5t$, where $t \ge 0$ represents the time elapsed in months. It is known that, after $k$ months from the start ($t=0$), the accumulated volume of cancellations reached a total of $3200$ customers (that is, $32$ hundreds of customers). a) Set up the equation that relates the definite integral $\int_0^k (6 - 0,5t)\,dt$ to the accumulated total and calculate the integrated expression as a function of the upper limit $k$ applying Barrow's rule. b) Solve the resulting quadratic algebraic equation to find the two possible mathematical values of $k$. c) Explain which of the two values must be discarded based on the economic meaning of the problem and the domain in which the cancellation rate is physically meaningful ($f(t) \ge 0$).

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 $0$.

1. a) Determine the values of the parameters $a$, $b$ and $c$.
2. b) Calculate the area of the region bounded by the graph of the function $f(x)$ and the x-axis $OX$.

A logistics company has modeled the inflow of packages to its central warehouse using the function $f(x) = |x^2 - 4x|$, where $x$ represents the hours elapsed from opening ($x=0$) to closing ($x=5$).

1. a) Express the function $f(x)$ as a piecewise function.

1. b) Calculate the total volume of packages received during the 5 hours, knowing that this value coincides with the definite integral of $f(x)$ on the interval $[0, 5]$.

1. c) Graph the function $f(x)$ on the interval $[0, 5]$ and shade the area associated with the calculation in part b).

Considera las funciones $f(x) = x^3 - 3x$ y $g(x) = x$. En la figura se muestran sus gráficas.

1. a) Find all the points where the two functions intersect.

1. b) Calculate the total area of the finite regions bounded by the graphs of $f(x)$ and $g(x)$. Keep in mind that the functions may change their relative position (which one is above) in the different intervals you find.

The rate of change of net profit of a tech start-up (in thousands of euros per month) is given by the function $B'(t) = 20 - 10e^{-0,2t}$, where $t$ is the time in months since the launch of its flagship product.

1. a) If at the time of the launch ($t=0$) the accumulated profit was $0$ euros, obtain the expression for the accumulated profit function $B(t)$.
2. b) Calculate, using Barrow's rule, the total net profit made by the company between month 5 and month 10. Express the result in euros, rounding to the nearest integer.

Let $f(x) = \frac{k}{x^2}$ be the function defined for $x > 0$. It is known that the area of the region bounded by the graph of this function, the $OX$-axis and the vertical lines $x = 1$ and $x = k$ (with $k > 1$) is equal to $0,75$ square units.

1. a) Calculate, with justification, the value of the positive parameter $k$.
2. b) For the value of $k$ found, determine the equation of the tangent line to the curve at $x = 2$.

Given the integral function $F(x) = \int_{1}^{x^2} \ln(t) \, dt$, defined for $x \geq 1$:

1. a) Use the Fundamental Theorem of Calculus and the Chain Rule to find the derivative function $F'(x)$.
2. 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.
3. c) Calculate the exact value of $F(e)$ by integrating by parts.

It is desired to design the profile of a skatepark ramp. The upper edge follows the curve $f(x) = e^{x/2}$ and the lower edge is bounded by the tangent line to said curve at the point with x-coordinate $x = 0$. The region is bounded on the right by the vertical line $x = 2$.

1. a) Find the equation of the aforementioned tangent line.
2. b) Calculate the cross-sectional area of the ramp (the region bounded by the curve, the tangent line and the vertical line).

Calculate the area of the shaded region in the figure, which represents the logo of a new fashion brand. The logo is bounded by the function $f(x) = \sqrt{x+4}$, the $OY$ axis, and the line $y = \frac{1}{2}x + 1$.

Note: You must first find the intersection point of both functions in the first quadrant in order to set the limits of integration correctly.

Consider the integral function $F(x) = \int_2^x (t^3 - 6t^2 + 9t)\,dt$, defined for all $x \in \mathbb{R}$. To determine the intervals of monotonicity and the relative extrema of $F(x)$, two students propose two different methodological approaches: Method A consists of first finding the explicit expression for $F(x)$ by determining an antiderivative and integrating using Barrow's rule, and then differentiating with respect to $x$; Method B proposes applying the Fundamental Theorem of Calculus directly to write $F'(x)$ without calculating any prior antiderivative. a) Fully carry out Method A (calculating the antiderivative, applying Barrow's rule on $[2, x]$, and differentiating the resulting polynomial) and verify that it matches the immediate result provided by Method B. b) Compare both approaches: which one is more efficient? Justify why Method A would be unfeasible for a Bachillerato student if the function inside the integral were $g(t) = e^{-t^2}$, whereas Method B would still allow the problem to be solved. c) Using the derivative $F'(x)$, determine the critical points of $F(x)$, its intervals of increase and decrease, and classify its relative extrema with justification.

The sorting centre of an e-commerce company in Spain monitors the outbound rate of orders over an 8-hour shift using the following function $r(t)$ (expressed in hundreds of parcels per hour, with $t \in [0, 8]$): $$r(t) = \begin{cases} a t^2 + b & \text{if } 0 \le t \le 2 \\ 10 - (t - 4)^2 & \text{if } 2 < t \le 8 \end{cases}$$ In order for the changes in pace not to cause operational breakdowns and for it to be permissible to apply Barrow's rule piecewise over the entire interval, the function $r(t)$ must be continuous. a) State and deduce the exact algebraic condition that the constants $a$ and $b$ must satisfy to ensure the continuity of $r(t)$ at the transition point $t = 2$. b) It has been verified that during the first two hours of the shift ($t \in [0, 2]$), an accumulated total of $800$ parcels ($8$ hundreds) were sorted. Set up the definite integral corresponding to that interval, solve it using Barrow's rule, and calculate the values of the parameters $a$ and $b$. c) Using the values found, calculate the total number of parcels sorted by the end of the full 8-hour shift using the interval additivity property, and determine at which specific hour of the shift the peak sorting rate was reached.