Exercises: Area under a curve

An athlete is training for a marathon. Their speed over a 15-minute stretch is constant and given by the function $v(t) = 12$ km/h, where $t$ is time in hours. Use the definite integral to calculate the distance travelled (the area under the speed curve) between the times $t = 0$ and $t = 0,25$ hours. Check the result using the geometric formula for the area of a rectangle.

An e-scooter rental company has found that the rate of energy consumption of one of its models follows the function $f(x) = 2x + 1$, where $x$ represents the kilometres travelled. Calculate the area enclosed by the graph of $f(x)$, the $OX$-axis and the vertical lines $x = 0$ and $x = 3$. What geometric shape does this region correspond to?

Calculate the area of the plane region bounded by the parabola $f(x) = x^2 + 2$, the x-axis and the lines $x = 0$ and $x = 2$. Since the function is always positive in that interval, set up the definite integral directly using Barrow's rule.

The marketing department of a new music app has determined that the marginal revenue per thousand new subscribers is $I'(x) = 10 - x$ (in hundreds of euros), where $x$ is the number of thousands of subscribers. Calculate the total accumulated profit (area under the curve) when the number of subscribers increases from $x = 1$ thousand to $x = 4$ thousand.

Find the area of the region bounded by the constant function $f(x) = 4$, the $OX$-axis and the lines $x = -1$ and $x = 2$. Justify the result graphically by calculating the area of the resulting rectangle without using integrals.

Calculate the area of the shaded region in the following figure, given that the curve corresponds to the function $f(x) = \sqrt{x}$ and is bounded by the $OX$-axis and the vertical line $x = 4$.

Given the polynomial function $f(x) = 3x^2 + 2x$, find the area bounded by its graph and the $OX$-axis in the interval $[1, 2]$. Check that, in this interval, the function does not cross the $OX$-axis (that is, it is always positive), which facilitates direct calculation.

A graphic designer is creating a logo that includes a shape bounded by the function $f(x) = e^x$. Calculate the area of the part of the logo that lies between the $OX$-axis, the curve itself, and the lines $x = 0$ and $x = 1$. (Remember that the integral of $e^x$ is itself).

Marcos has to calculate the area of the plane region bounded by the curve $f(x) = x^2 - 2x$, the $x$-axis, and the vertical lines $x = 0$ and $x = 3$. On his exam, he writes the following:

“$\text{Area} = \int_0^3 (x^2 - 2x)\,dx = \left[\frac{x^3}{3} - x^2\right]_0^3 = \left(\frac{27}{3} - 9\right) - 0 = 0\text{ u}^2$”.

Upon obtaining zero, he concludes that the function does not enclose any area with the axis on that interval.

1. a) Identify, with reasoning, the conceptual error made by Marcos and explain why a definite integral can equal zero while the enclosed geometric area is strictly positive.
2. b) Correct Marcos's procedure: find the $x$-intercepts of the function, split the interval into the corresponding regions, and calculate the exact value of the actual area.

The entrance to an underground ventilation tunnel has a parabolic profile modelled by the function $f(x) = 9 - x^2$ (where $x$ and $f(x)$ are measured in metres), bounded below by the horizontal ground ($y = 0$). You want to determine the cross-sectional area of this entrance.

1. a) Before integrating, make a reasoned estimate of the area by bounding it between two numerical values: find the area of the rectangle that circumscribes the parabola (using the intersection points with the ground and the height of the vertex) and the area of an inscribed isosceles triangle with that same base and height. Justify why the actual area must lie within that interval.
2. b) Calculate the exact area using the definite integral by applying Barrow's rule.
3. c) Check whether the result obtained confirms your previous estimate and determine what exact percentage of the area of the circumscribed rectangle the parabola occupies.

A skatepark designer wants to calculate the area of the side section of a new ramp. The profile of the ramp follows the curve of the function $f(x) = \frac{1}{4}x^2 + 1$ in the interval $[0, 4]$, where $x$ and $y$ are measured in metres. Calculate the area of the side wall between the ramp, the x-axis ($OX$) and the vertical lines $x=0$ and $x=4$.

Calculate the area of the region in the plane enclosed between the graph of the function $f(x) = -x^2 + 4x$ and the line $g(x) = x$. To do this, first identify the points of intersection between both functions and sketch the region.

A solar energy company has determined that the monthly savings rate (in thousands of euros) generated by one of its panels after installation is given by the function $S(t) = t^2 - 6t + 8$, where $t$ is the time in months.

1. a) Determine in which months the savings are negative (maintenance costs exceed savings).
2. b) Calculate the total net savings (total area above the axis minus the area below the axis) during the first 4 months.
3. c) Calculate the total physical area enclosed between the graph and the $OX$ axis on the interval $[0, 4]$.

Find the area of the region bounded by the curve $f(x) = x^3 - 4x$ and the x-axis. Take into account that the function can take positive and negative values, so you will need to calculate the area of each region separately and sum their absolute values.

Consider the marginal profit function of a mobile app store $B'(x) = 100 - 2x$, where $x$ is the number of daily subscribers.

1. a) Calculate the profit function $B(x)$ knowing that for 0 subscribers the profit is $-500$ euros (fixed costs).
2. b) Use the definite integral to calculate the total increase in profit when the number of subscribers increases from 20 to 40.

Find the value of a positive parameter $k$ such that the area of the region bounded by the function $f(x) = kx^2$, the $OX$-axis and the vertical line $x=3$ is equal to 18 square units.

Given the piecewise-defined function:

$f(x) = \begin{cases} 2x & \text{if } 0 \le x \le 1 \\ 3 - x & \text{if } 1 < x \le 3 \end{cases}$

Calculate the area of the region bounded by the graph of $f(x)$ and the x-axis in the interval $[0, 3]$.

Calculate the area of the region bounded by the graphs of the functions $f(x) = e^x$ and $g(x) = e^{-x}$ and the vertical line $x = 1$. Sketch the region before carrying out the calculation.

You can do it on paper and send a photo with your answer.

On an irrigated farm in the Region of Murcia, a photovoltaic solar installation supplies power to the water pumps. On a clear summer day, the instantaneous electrical power supplied is modelled by the function $P(t) = -2(t - 14)^2 + 50$, where $t$ represents the solar hour of the day ($t \in [9, 19]$) and $P(t)$ the power in kilowatts ($\text{kW}$). The total accumulated energy (in $\text{kWh}$) coincides with the area under the power curve.

1. a) Interpret the physical and temporal meaning of the coordinates of the parabola's vertex in the context of the farm.
2. b) Set up and calculate the definite integral that gives the total accumulated energy during the central hours of the day, between $11:00$ and $17:00$ hours ($t = 11$ and $t = 17$).
3. c) If the cost of electricity from the grid during that peak period is $0,18$ € per $\text{kWh}$, calculate the total economic savings generated by solar pumping in that six-hour interval.

An architect designs the side profile of an open-air grandstand for a municipal auditorium. The structure is bounded by the ground ($x\text{-axis}$) and the curve $f(x) = k \cdot (6x - x^2)$ on the interval $[0, 6]$, where $x$ and $y$ are in metres and $k > 0$ is a height constant to be determined.

1. a) Set up the integral equation representing the lateral area under the curve in terms of $k$ and obtain the exact value of this parameter so that the section covers exactly $72\text{ m}^2$.
2. b) With the value of $k$ found, it is planned to cover the first half of the auditorium (an area of $36\text{ m}^2$) with a waterproof tarp, from $x = 0$ to a vertical line $x = c$. Without needing to set up any additional integral, deduce and justify the value of $c$ using the symmetry properties of the parabola.
3. c) If the tarp were to cover only a quarter of the total area ($18\text{ m}^2$) from $x = 0$ to $x = c_1$, set up the polynomial equation that $c_1$ should satisfy and explain, without solving it, why this solution must strictly satisfy $0 < c_1 < 3$.

A new logo is being designed for a financial technology company. The basis of the design is bounded by the graphs of the functions $f(x) = -x^2 + 6x$ and $g(x) = 2x$, which are shown in the figure.

1. a) Determine the points of intersection of both functions.
2. b) Calculate the area of the region enclosed by both graphs, which will represent the main area of the logo.
3. c) If it is decided to add a vertical line $x = k$ (with $0 < k < 4$) that divides this region into two exactly equal areas, set up the integral equation that would allow you to find the value of $k$.

A streaming platform has analysed that the rate of growth of new daily subscribers (in thousands) follows the function $f(t) = t \cdot e^{-t/4}$, where $t$ is the time in days since the release of a hit series.

Calculate the total number of accumulated subscribers during the first 8 days. To do this, you will need to find the area under the curve $f(t)$ in the interval $[0, 8]$. Note: Use integration by parts.

Consider the rational function $f(x) = \frac{2x + 5}{x + 1}$ defined for $x > -1$.

1. a) Find the area of the region bounded by the graph of $f(x)$, the $OX$-axis and the vertical lines $x = 0$ and $x = 3$.
2. b) Determine the equation of the tangent line to the curve at the point with x-coordinate $x = 0$.
3. c) Calculate the area of the triangle formed by this tangent line, the $OX$-axis and the $OY$-axis, and compare it with the area obtained in part (a).

An architect wants to design a window with an inverted parabolic shape given by the function $f(x) = 4 - x^2$. To provide stability, a horizontal transom will be placed along the line $y = a$ (with $0 \le a < 4$).

Find the value of the constant $a$ so that the area of the window region above the transom is exactly $\frac{9}{2}$ square units.

The marginal revenue function of an agricultural cooperative is given by $I'(x) = 100 - 0,2x^2$ (in euros per unit sold), where $x$ represents the number of tonnes of product.

1. a) Calculate the total revenue function $I(x)$ knowing that for zero sales, revenue is zero.
2. b) Determine the area enclosed by the curve $I'(x)$ and the x-axis in the interval where the marginal revenue is positive. Explain the economic meaning of this value.

Find the area of the region bounded by the function $f(x) = |x^2 - 4|$, the x-axis and the lines $x = -1$ and $x = 3$.

Tip: Identify the intervals where the expression inside the absolute value changes sign and split the definite integral into the sum of several integrals accordingly.

Given the parabola $f(x) = x^2 - 4x + 3$, whose graph is shown below:

1. a) Find the equation of the tangent line to the parabola at the point where it cuts the $OY$ axis.

1. b) Calculate the area of the finite region bounded by the parabola, the tangent line found in part a), and the $OX$ axis.

Calculate the area of the region in the first quadrant bounded by the function $f(x) = \ln(x)$, the y-axis $OY$, the x-axis $OX$ and the horizontal line $y = 2$.

Note: It may be helpful to calculate the area by integrating with respect to the y-axis or, alternatively, by subtracting areas of known rectangles.

You want to calculate the area of the planar region located in the first quadrant that is simultaneously bounded by the curve $y = x^2$, the line $y = 6 - x$, and the abscissa axis ($y = 0$).

1. a) Analyse and compare the following three possible strategies to solve the problem:

• Strategy 1: Find the intersection of both functions, split the region into two vertical sections with respect to the $x$-axis, and sum two definite integrals.

• Strategy 2: Solve for $x$ in terms of $y$ in both equations and integrate with respect to the $y$-axis using a single definite integral.

• Strategy 3: Integrate the parabola up to the point of intersection with the line and add the geometric area of the triangle formed by the line with the $x$-axis, without integrating the line.

Justify which of the three strategies is the most efficient and carries the lowest risk of algebraic error.

1. b) Solve the problem applying the strategy you selected as optimal.
2. c) A student claims that the requested area can be obtained directly by calculating $\int_0^6 ((6 - x) - x^2)\,dx$. Explain with analytical and geometric rigor the double conceptual error contained in this proposal and what region of the plane that integral would actually represent.

In economic sociology, the Lorenz curve $L(x)$ describes the income distribution of a country, where $x \in [0, 1]$ represents the cumulative proportion of the population ordered from lowest to highest income, and $L(x) \in [0, 1]$ the cumulative proportion of total income that this population receives. The line $y = x$ represents perfect equality. The Gini index, which measures inequality, is defined by the expression $G = 2 \cdot \int_0^1 (x - L(x))\,dx$, so that $G \in [0, 1]$ (where $0$ is absolute equality and $1$ is extreme inequality).

A study by the National Statistics Institute (INE) models the pre-tax income of a region using the curve $L(x) = \frac{1}{2}x^2 + \frac{1}{2}x^3$.

1. a) Verify analytically that the function satisfies the boundary conditions of every Lorenz curve ($L(0) = 0$ and $L(1) = 1$) and calculate the initial Gini index of the region.
2. b) To combat the social gap, the administration implements a tax reform with direct transfers, projecting that the new Lorenz curve takes the power form $L_k(x) = x^k$ with $k > 1$. Set up the corresponding integral equation and determine the exact value of the parameter $k$ required for the Gini index to decrease to $0,25$.
3. c) Interpret what would happen to the area enclosed between the curve and the line of equality if the parameter $k$ were to progressively approach $1$, and explain what economic implication this behaviour would have on the distribution of wealth.