Area under a curve

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.

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 $2\text{ u}^2$.

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:

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

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 $x = 2$.

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 $x = 9$.

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.

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.

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

Aoife 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 Aoife'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.

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.

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.

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.

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 $x = 3$.

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.

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.

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.

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.

According to solar radiation records from Met Éireann at a meteorological station in Dublin 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 $\int_0^{24} P(t)\,dt$.

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 ($Eje\ OX$).