Exercises: Area under a curve

Consider the linear function $f(x) = \frac{1}{2}x + 1$. You are required 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 formulas (area of a trapezoid).

2. Calculate the definite integral $\int_{0}^{4} (\frac{1}{2}x + 1) \, dx$ by applying the Fundamental Theorem of Calculus 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, find the result of:

A car wash has a metal 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 structure of the parabola and the ground ($OX$-axis), given that the support points on the ground are $x = -2$ and $x = 2$.

In physics, the distance traveled by an object can be calculated as the area under the curve on a velocity-time graph. If an electric skateboarder moves at a constant speed of $v(t) = 12$ mph during the first half hour ($t = 0,5$ h), set up and calculate the definite integral that represents the distance traveled 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 ($OX$).

Hint: First you must find the points of intersection of the function with the $OX$-axis to determine the limits of integration. Remember that, if the function is negative in the interval, the area is the absolute value of the integral.

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, showing your reasoning, 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: Observe that the function intersects the $OX$-axis at a point within the given interval, so you will have 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 $x = 9$.

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

Determine the area of the region bounded by the curve $f(x) = x^3 - 4x$ and the $OX$-axis in the interval $[-1, 2]$. Bear in mind 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 the absolute value implies that the function is always non-negative, but its analytical expression changes at the points where the argument is zero.

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

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 to a tunnel whose front section can be modeled by the function $f(x) = 4 - \frac{1}{4}x^2$, where $x$ and $y$ are expressed in feet. The floor of the tunnel coincides with the $OX$-axis. Calculate the area of the cross-section 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.

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 following 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 aforementioned 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 $k$ so that the area of this region is exactly $4$ square units.

The function $f(x) = |x^2 - 4|$ is defined. We want to study the region of the plane that is "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 absolute value function to split the integral if necessary.

Consider the function $f(x) = \ln(x)$. Find:

1. Find the equation of the normal line to the graph of $f(x)$ at the point $P(1, 0)$.

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 area of the cross-section of the tank by calculating the definite integral in each interval.

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

Hint: You must first locate the relative maximum using the derivative, find the value of the y-coordinate at this 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 areas.

Calculate the exact value of the constant $c$.