Exercises: Area bounded by two curves

A graphic designer is creating a new logo for an urban clothing brand. The central design consists of the area enclosed between the parabola $f(x) = -x^2 + 4$ and the line $g(x) = x + 2$, which you can see in the figure.

To find out how much silver paint will be needed for the print, solve:

1. a) Find the points of intersection between both functions.

1. b) Calculate the area of the region enclosed between the two graphs (in square units).

In an extreme sports park, a skate ramp is to be installed. The side profile of the ramp is defined by the curve $f(x) = x^2$ and is to be covered with a safety tarp reaching the horizontal platform defined by the line $g(x) = 4$.

Calculate the area of the side section between the ramp and the horizontal platform (units in metres).

Two tech companies are comparing their user growth during their first year. The number of users (in thousands) of Company A follows the function $f(x) = x^2$, while Company B follows the function $g(x) = x$.

Calculate the area of the region between both graphs in the interval $[0, 1]$ to visualise the total accumulated difference in users during that first year.

A solar panel with a special geometric shape is being designed. The edges of the panel are defined by the parabola $f(x) = 2x - x^2$ and the inverted parabola $g(x) = x^2 - 2x$. The figure shows both curves and the region that forms the panel.

Calculate the intersection points of both curves and the total surface area of the solar panel.

An architect wants to design a window for a modern building. The top part is bounded by the curve $f(x) = 6x - x^2$ and the bottom part by the line $g(x) = x$.

Determine the region enclosed by these two functions and calculate its area to estimate the amount of glass needed.

In an environmental engineering project, the cross section of an irrigation channel is studied. The profile of the channel is given by the curve $f(x) = x^2 - 1$ and the maximum water level is bounded by the line $g(x) = 3$.

Calculate the area of the channel's cross section that lies below the water level.

Calculate the area of the plane region bounded by the functions $f(x) = \sqrt{x}$ and $g(x) = x^2$ in the first quadrant.

Hint: Remember that $\sqrt{x}$ can be expressed as a power with a fractional exponent to facilitate integration.

Determine the area of the region bounded by the line $y = 2x$ and the curve $y = x^2 - 3$.

To do so, first find the points of intersection by setting both functions equal to each other and apply Barrow's rule to the integrand of the difference function.

Lucía has to calculate the area of the bounded region enclosed between the graphs of $f(x) = 4 - x^2$ and $g(x) = 2 - x$. On her exam, she writes the following:

1. “Points of intersection: I set $f(x) = 0 \implies 4 - x^2 = 0 \implies x = -2$ and $x = 2$, then I integrate between $-2$ and $2$”.

2. “Relative position: I take $x = 0$; since $f(0) = 4$ and $g(0) = 2$, the parabola is above the line”.

3. “Calculation: $\text{Area} = \int_{-2}^{2} [(4 - x^2) - (2 - x)]\,dx = \left[ 2x + \frac{x^2}{2} - \frac{x^3}{3} \right]_{-2}^{2} = \frac{8}{3} \approx 2,67$ square units”.

1. a) Identify and explain in a reasoned manner the two conceptual errors that Lucía made in her approach.
2. b) Correct the exercise by solving it step by step: find the correct limits of integration, check the relative position on the appropriate interval, and calculate the exact value of the area.

We want to calculate the area of the region enclosed between the parabola $f(x) = -x^2 + 4x$ and the line $g(x) = x$.

1. a) Find the points of intersection between both curves and determine the coordinates of the vertex of the parabola.
2. b) Estimate before integrating: Consider the rectangle whose base is the segment between the intersection abscissae and whose height is the maximum value of $f(x)$, and also consider the triangle that connects the two intersection points with the vertex of the parabola. Justify, using geometric reasoning, why the desired area must be strictly less than that of the rectangle and greater than that of the triangle, indicating the numerical interval in which the result must lie.
3. c) Check: Calculate the exact value of the area using the definite integral and verify whether the obtained value is consistent with your previous estimation.

Calculate the area of the plane region enclosed between the graphs of the functions $f(x) = -x^2 + 6x$ and $g(x) = x$. Start by identifying the points of intersection of both curves and make a sketch of the region before applying Barrow's rule.

Find the area of the region bounded by the graphs of the functions $f(x) = x^2 - 2x + 1$ and $g(x) = -x^2 + 4x + 1$. Determine which of the two functions lies above the other in the interval of integration by analysing the sign of the difference function $h(x) = f(x) - g(x)$.

A design company wants to create a new logo using the region enclosed by the curves $f(x) = \sqrt{x}$ and $g(x) = x^2$. Calculate the area of this region (expressed in square units) in order to budget the cost of printing it on perspex.

Given the function $f(x) = x^2 - 4$, find the equation of the tangent line to its graph at the point with x-coordinate $x = 3$. Once obtained, calculate the area of the region bounded by the graph of $f(x)$, the tangent line found, and the y-axis ($OY$).

Determine the area of the region enclosed by the cubic function $f(x) = x^3 - 4x$ and the line $g(x) = 0$ (the x-axis). Keep in mind that the region consists of two separate areas and that you will need to calculate the area of each one separately using absolute values if necessary.

The profile of a bicycle ramp in an urban park follows the curve $f(x) = \frac{x^2}{10} + 2$. It has been decided to cover the gap underneath between the ramp and a straight support structure connecting the points $A(0, 2)$ and $B(10, 12)$. Calculate the area of the side section enclosed between the curved ramp and the straight structure.

Find 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 functions, indicating the point where they cross.

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

Calculate the value of the positive real parameter $k$ such that the area of the region bounded by the parabola $y = x^2$ and the line $y = kx$ is equal to $36$ square units. Hint: Express the area in terms of $k$ and equate the result to the given value to solve for the unknown.

An irrigation community in the province of Seville has a photovoltaic solar facility to power its water pumps. On a sunny spring day, the electrical power generated (in kilowatts, $\text{kW}$) between 8:00 ($t = 0$) and 18:00 ($t = 10$) is modeled by the function $P(t) = -t^2 + 10t$. On the other hand, the continuous consumption required by the pumping system follows the model $C(t) = 2t + 7$ (in $\text{kW}$).

1. a) Determine the hours of the day (values of $t$) between which the production of the panels exceeds the demand of the pumps, generating an energy surplus.
2. b) Set up and solve the definite integral that gives the total surplus energy (in kilowatt-hours, $\text{kWh}$) produced throughout that day.
3. c) If the community feeds this surplus into the electrical grid and is compensated at a price of $0,08$ euros per $\text{kWh}$, what is the economic revenue obtained that day thanks to the surplus?

Consider the parabola $f(x) = x^2$ and its tangent line at the point with abscissa $x = 2$, which we will call $r$. We want to determine the area of the bounded region enclosed between the curve $f(x)$, the tangent line $r$, and the x-axis ($y = 0$).

1. a) Obtain the equation of the tangent line $r$ and calculate its intersection point with the x-axis.
2. b) To calculate the area of this region, two methods can be followed:

• Method 1 (horizontal decomposition): Divide the region with a vertical line into two subintervals and add two definite integrals.

• Method 2 (geometric subtraction): Calculate the total area under the parabola on the interval $[0, 2]$ and subtract the area of the triangle formed by the tangent line, the x-axis, and the vertical line $x = 2$.

Explain clearly why Method 1 requires using two different integrals and why Method 2 is considerably more direct and carries a lower risk of calculation error.

1. c) Apply Method 2 to find the exact area of the region and check that it matches the formulation of Method 1.

A graphic design company is creating a new logo that includes an aerodynamic shape. The upper part of the logo follows the path of the function $f(x) = -x^2 + 6x - 5$. It has been decided to add an aesthetic line that is the tangent line to the graph of $f(x)$ at the point with x-coordinate $x = 4$. Calculate the area of the region in the first quadrant bounded by the curve $f(x)$, the aforementioned tangent line, and the y-axis ($eje \: OY$).

Consider two functions representing the airflow in a wind tunnel during a 10-second experiment: $f(x) = x^3 - 6x^2 + 9x$ and $g(x) = x^2 - 3x$. The engineers need to know the total "interference volume", which is equivalent to the area enclosed between both graphs across their entire domain. Find the points of intersection between the two functions and calculate the total area of the regions they bound, taking into account that the functions may switch positions (which one is above the other) in the different intervals.

The cross section of an irrigation channel is being designed. The bottom of the channel is defined by the parabola $f(x) = \frac{1}{2}x^2$, while the water surface on a day of maximum flow rate is bounded by the curve $g(x) = \sqrt{2x}$. Determine the value of a constant $k$ such that the vertical line $x = k$ divides the area of the water's cross section into two equal parts.

An architect is designing a decorative window whose silhouette is bounded by the modulus function $f(x) = |x^2 - 4|$ and the horizontal line $g(x) = 2$. Due to the complexity of the shape, the glass must be cut into several pieces. Calculate the total area of the window, correctly identifying the limits of integration and the functions acting as upper and lower boundaries in each section.

In a microeconomics study, the supply function for a product is $S(x) = x^2 + 2x + 1$ and the demand function is $D(x) = 11 - x^2$, where $x$ is the number of units in thousands. "Consumer Surplus" is defined as the area bounded by the demand curve, the horizontal line passing through the equilibrium point (where supply equals demand), and the y-axis. Calculate this surplus by first finding the equilibrium point in the first quadrant.

Find the area of the region bounded by the curve $f(x) = e^{x-1}$, the tangent line to that curve at the point where its gradient is equal to $1$, and the vertical line $x = 0$.

Note: Remember that to find the point of tangency you must first solve $f'(x) = 1$.

Consider the function $f(x) = x^3 - 3x^2 + 4$. Calculate the area of the finite region bounded by the graph of this function and the line passing through its two stationary points (the relative maximum and the relative minimum). You must first find the coordinates of these points and the equation of the line joining them.

The figure shows the graph of $f$, the line $r$ and the regions bounded by both.

Determine the area of the region enclosed between the graphs of the functions $f(x) = \frac{8}{x}$ and $g(x) = 6 - x$. In this case, the region is entirely located in the first quadrant. Identify the intersection points by solving the resulting equation and justify which of the two functions lies above the other on the interval of integration before applying Barrow's rule.

Consider the bounded region limited by the parabola $f(x) = 2x - x^2$ and the x-axis ($y = 0$). A line passing through the origin of coordinates is drawn, with equation $y = mx$ (with $0 < m < 2$), which divides this region into two areas: an upper one (between the parabola and the line) and a lower one (between the line and the $OX$ axis).

1. a) Calculate the total area enclosed between the parabola and the x-axis.
2. b) Determine, in terms of the slope $m$, the abscissa of the non-zero intersection point between the parabola and the line, and obtain the algebraic expression (in terms of $m$) of the area of the upper region.
3. c) Find the exact value of the slope $m$ so that the line divides the original region into two parts of exactly equal area.
4. d) The line connecting the origin $(0,0)$ to the vertex of the parabola has slope $m = 1$. Without carrying out laborious calculations, explain geometrically why the slope $m$ obtained in the previous section must be strictly less than $1$, considering how the area mass of the parabola is distributed across the interval.

The mobility department of a city council analyzes the hourly inflow of travelers (in hundreds of people per hour) at two entrances of a central station during the 4 peak hours ($t \in [0, 4]$). These flows are modeled by the functions:

A preliminary report states: "Since $\int_0^4 [f(t) - g(t)]\,dt = 0$, both entrances operated with identical overall net influx and, therefore, there is no discrepancy or enclosed area between both flow profiles."

1. a) Find all instants $t \in [0, 4]$ at which both entrances record exactly the same flow of travelers.
2. b) Refute the report's conclusion: explain in detail why a zero integral does not imply that the area between the curves is zero, distinguishing between net accumulated balance and geometric area between curves.
3. c) Calculate the real total area enclosed between both functions on the interval $[0, 4]$ and interpret the meaning of each of the resulting regions in terms of which entrance bears a greater influx in each section.
4. d) Reason, without computing a new antiderivative, what value the integral $\int_0^4 [f(t) - g(t)]\,dt$ would take if the flow $f(t)$ were uniformly increased by $1$ hundred travelers/hour throughout the entire period. Would the geometric area enclosed between both curves also change? Justify your answer.

Topic challenge. How does the energy transition impact daily consumption? Analyse electricity demand and renewable generation in your autonomous community using official data and integral calculus tools.

1. Data collection and modelling: Download yesterday's hourly demand and solar generation curves from the Red Eléctrica de España (REE) website. In GeoGebra or a spreadsheet, fit the demand to a continuous function and normalise it to define a probability density function $f(t)$ with $\int_{0}^{24} f(t)\,dt = 1$, calculating its antiderivative by integration by parts or partial fraction decomposition.

2. Calculation of areas and probabilities: Determine the probability that critical consumption occurs between $19,0$ and $22,0$ hours. Model the solar curve $S(t)$ with a semi-immediate integral and apply Barrow's rule to calculate the total area under the demand curve (energy consumed) and the area between demand and solar generation (energy not covered by photovoltaics).

3. Executive report: Submit a one-page report with the overlaid graphs, the step-by-step analytical development of each integral, and a reasoned proposal on energy storage.

Assessment criteria:

• Algebraic rigour in calculating antiderivatives (immediate, rational, and by parts) and accurate application of Barrow's rule.

• Proper mathematical processing and modelling of real REE data using GeoGebra or a spreadsheet.

• Correct probabilistic interpretation of the areas and contextualised justification of the area between both curves.