Exercises: Area under a curve

Consider the linear function $f(x) = \frac{1}{2}x + 1$. You are asked to calculate the area of the shaded region in the graph, bounded by the function, the $OX$-axis, and the vertical lines $x = 0$ and $x = 4$. To do this:

1. Calculate the value of the area using elementary geometric formulae (area of a trapezium).

2. Calculate the definite integral $\int_{0}^{4} (\frac{1}{2}x + 1) \, dx$ using Barrow's rule and check that both results coincide.

Calculate the value of the definite integral of the polynomial function $f(x) = 3x^2 - 2x + 5$ in the interval $[1, 2]$. That is, obtain the result of:

A car wash tunnel has a metallic structure whose upper part follows the shape of the parabola $f(x) = -x^2 + 4$. Calculate the area of the flat (front) surface bounded by the parabolic structure and the ground ($OX$-axis), knowing that the support points on the ground are $x = -2$ and $x = 2$.

In physics, the distance travelled by an object can be calculated as the area under the curve in a velocity against time graph. If an e-scooter moves at a constant velocity of $v(t) = 12$ km/h during the first half hour ($t = 0,5$ h), set up and calculate the definite integral that represents the distance travelled in that time.

Find the area of the plane region bounded by the graph of the function $f(x) = x^2 - 9$ and the x-axis ($OX$).

Use the properties of the definite integral to solve the following problem:

Given that $\int_{1}^{4} f(x) \, dx = 7$ and that $\int_{4}^{6} f(x) \, dx = -3$, calculate, with justification, the value of:

1. $\int_{1}^{6} f(x) \, dx$

2. $\int_{4}^{1} 2 \cdot f(x) \, dx$

Calculate the total area of the region bounded by the line $f(x) = x - 2$, the $OX$-axis, and the vertical lines $x = 0$ and $x = 3$.

Note: Notice that the function crosses the $OX$-axis at a point inside the given interval, so you will need to split the region into two sub-areas.

A graphic design company wants to create a logo that includes a curved shape defined by $f(x) = \sqrt{x}$. Calculate the area of the part of the logo enclosed between this curve, the $OX$-axis, and the vertical line $x = 9$.

Lucía needs to calculate the area of the region bounded by the curve $f(x) = x^2 - 1$, the $x$-axis, and the vertical lines $x = 0$ and $x = 2$. On her exam, she writes the following working:

1. a) Without carrying out any new calculations yet, explain in detail Lucía's conceptual error based on the sign of $f(x)$ on the interval $[0, 2]$, and explain why her final result is smaller than the actual area.
2. b) Correct the exercise by properly dividing the integration interval, and calculate the exact value of the area.

You want to calculate the area of the closed region bounded by the parabola $f(x) = 4 - x^2$ and the positive coordinate axes in the first quadrant.

1. a) Before integrating, make a preliminary estimate of the area by bounding it between two elementary geometric figures: the right-angled triangle with vertices $(0,0)$, $(2,0)$, and $(0,4)$, and the circumscribed rectangle of dimensions $2 \times 4$. Based on these two bounds, propose an estimated numerical value for the area.
2. b) Calculate the exact value of the area by applying Barrow's rule to the corresponding definite integral.
3. c) Check whether the exact value obtained lies within the bounds established in part a) and determine the absolute error made by your initial estimate relative to the true value.

Calculate the area of the plane region bounded by the graph of the function $f(x) = -x^2 + 6x - 5$ and the x-axis $OX$. Sketch the region before performing the calculations to identify the x-intercepts.

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

Determine the area of the region bounded by the curve $f(x) = x^3 - 4x$ and the $OX$-axis in the interval $[-1, 2]$. Note that the function may change sign within the interval, so you will need to split the integral if necessary.

A graphic design company wants to create a logo that includes the shape of a "wave". This shape is defined by the function $f(x) = \sin(x) + 1$ in the interval $[0, \pi]$. Calculate the total area of the figure bounded by the curve, the $OX$-axis, and the vertical lines $x = 0$ and $x = \pi$.

Find the area of the region bounded by the function $f(x) = |x^2 - 1|$ and the x-axis between $x = 0$ and $x = 2$. Remember that the definition of modulus implies that the function is always non-negative, but its algebraic expression changes at the points where the argument becomes zero.

Consider the function $f(x) = x^2 - 2x + 2$. Calculate the area of the region bounded by the graph of $f(x)$, the $x$-axis, the $y$-axis, and the tangent line to the curve at the point with abscissa $x = 3$.

Find the value of a positive parameter $k$, given that the area of the region bounded by the parabola $f(x) = kx^2$ and the $OX$-axis between $x = 0$ and $x = 3$ is exactly $18$ square units.

An architect is designing the entrance of a tunnel whose front cross-section can be modelled by the function $f(x) = 4 - \frac{1}{4}x^2$, where $x$ and $y$ are expressed in metres. The floor of the tunnel coincides with the $OX$-axis. Calculate the cross-sectional area of the tunnel entrance.

Calculate the area of the region bounded by the curve $f(x) = \sqrt{x}$, the $OX$-axis, and the line $y = -x + 6$ in the first quadrant. Hint: first identify the point of intersection between the curve and the line to determine the limits of integration.

Consider the family of curves given by $f_k(x) = kx - x^2$, where $k$ is a positive real parameter ($k > 0$).

1. a) Find the intersection points of the graph of $f_k$ with the $x$-axis and determine, in terms of the parameter $k$, the expression for the area of the region bounded by the curve and that axis.
2. b) Calculate the specific value of $k$ so that the area of that region is exactly equal to $36\text{ u}^2$.
3. c) For the value of $k$ obtained in the previous part, a vertical line of equation $x = c$ (with $0 < c < k$) is to be drawn to divide the region into two subregions of equal area. Without setting up any new integral equation, deduce and justify what the exact value of $c$ must be, relying on the symmetry properties of the function.

According to solar radiation records from AEMET at a meteorological station in Seville during a clear spring day, the instantaneous power generated by a photovoltaic installation is modelled by the function:

where $P(t)$ represents the power in megawatts ($\text{MW}$) and $t$ is the hour of the day measured in the interval $[0, 24]$.

1. a) Determine the interval of hours of the day $[t_1, t_2]$ during which the installation effectively generates power ($P(t) \ge 0$).
2. b) Knowing that the total accumulated energy produced (in $\text{MWh}$) is equal to the area under the power curve during the hours of production, calculate the total energy generated throughout that day.
3. c) If the energy is paid at a fixed rate of $50\text{ €/MWh}$, find the day's financial revenue. Explain why it would have been a methodological error to calculate the energy using the direct integral $\int_0^{24} P(t)\,dt$.

An architect is designing the side profile of a new sports hall. The main structure follows the curve of the function $f(x) = -x^2 + 6x$. To ensure stability, a metal reinforcement must be placed along the tangent line to the curve at the point with x-coordinate $x = 4$.

Calculate the area of the plane region bounded by the graph of the function $f(x)$, the mentioned tangent line, and the y-axis ($Eje\ OY$).

Consider the functions $f(x) = e^x$ and $g(x) = e^{-x}$. Let $R$ be the region bounded by the graphs of both functions and the vertical line $x = k$ (where $k > 0$).

Determine the value of the positive parameter $k$ such that the area of this region is exactly $4$ square units.

Consider the function $f(x) = |x^2 - 4|$. We want to study the region of the plane "trapped" between this curve and the horizontal line $y = 5$.

Calculate the area of the region bounded above by the line $y = 5$ and below by the graph of the function $f(x)$. Note: Take into account the points of intersection and the definition of the modulus function to split the integral if necessary.

Let the function be $f(x) = \ln(x)$. You are asked to:

1. Find the equation of the normal line to the graph of $f(x)$ at the point $P(e, 1)$.

2. Calculate the area of the plane region bounded by the curve $y = \ln(x)$, the normal line found in the previous section, and the x-axis ($Eje\ OX$).

A water tank has a cross-section whose base is defined by the $OX$-axis and its upper part by the piecewise function:

$f(x) = \begin{cases} \sqrt{x+4} & \text{if } -4 \leq x < 0 \\ 2 \cos\left(\frac{\pi x}{4}\right) & \text{if } 0 \leq x \leq 2 \end{cases}$

Calculate the total cross-sectional area of the tank by computing the definite integral in each interval.

Determine the value of the constant $a \in \mathbb{R}$ with $a > 1$, knowing that the area of the region bounded by the curve $y = \frac{1}{x^2}$, the $x$-axis, and the vertical lines $x = 1$ and $x = a$ is equal to twice the area of the region bounded by the same curve $y = \frac{1}{x^2}$, the $x$-axis, and the lines $x = a$ and $x = a^2$.

Find the area of the region bounded by the function $f(x) = x^3 - 3x^2 + 2x$ and the horizontal line passing through the local maximum of this function.

Hint: You must first locate the local maximum using the derivative, find the y-coordinate at that point, and then determine the points of intersection between the curve and the resulting line.

Let the region $A$ be the region in the first quadrant bounded by the parabola $y = x^2$ and the line $y = 9$. There is a vertical line with equation $x = c$ (with $0 < c < 3$) that divides the region $A$ into two subregions of identical area.

Calculate the exact value of the constant $c$.

To calculate the area of the region bounded by the graph of $f(x) = x^3 - 4x$, the $x$-axis, and the lines $x = -2$ and $x = 2$, three classmates propose different strategies:

• Marcos finds the roots of $f(x)$ in $[-2, 2]$ ($x = -2$, $x = 0$, $x = 2$), studies the sign in each subinterval, and sets up: $\text{Area} = \left|\int_{-2}^0 f(x)\,dx\right| + \left|\int_0^2 f(x)\,dx\right|$.

• Sara first analyzes the global properties of $f(x)$, verifies that it is an odd function, and states that the area can be calculated in a more direct and cleaner way as: $\text{Area} = 2 \int_0^2 (-f(x))\,dx$.

• Diego states that, since the general definition of the area under a curve is $\int_{-2}^2 |f(x)|\,dx$, it is enough to directly find an algebraic antiderivative of $|x^3 - 4x|$ and apply Barrow's rule between $-2$ and $2$ without splitting it into pieces.

1. a) Evaluate the mathematical validity of the three procedures: explain why Diego's approach is not workable in practice and calculate the exact area using both Marcos's and Sara's methods to verify that both are correct.
2. b) Justify which of the two valid methods is more efficient and minimizes the risk of making sign errors with Barrow's rule.
3. c) If the graph is now shifted vertically by considering the function $g(x) = x^3 - 4x + 1$, justify whether Sara's symmetry shortcut could still be applied to calculate the area between the curve $g(x)$ and the $x$-axis on the interval $[-2, 2]$.

Consider the function $f_m(x) = x^2 - m^2$, where $m > 0$ is a constant real parameter. We wish to study the area $A(m)$ of the planar region bounded by the graph of $f_m(x)$, the $x$-axis, and the fixed vertical lines $x = 0$ and $x = 2$.

1. a) Note that the position of the positive root of the function relative to the interval $[0, 2]$ depends on the value of $m$. Reasonably deduce the analytical expression of the area function $A(m)$, distinguishing two cases: when $m \ge 2$ and when $0 < m < 2$.
2. b) Prove that the obtained function $A(m)$ is continuous at the boundary point $m = 2$.
3. c) Determine the exact value of $m \in (0, 2)$ for which the area of the region is exactly equal to $2\text{ u}^2$.