Exercises: Applications of the Definite Integral

Calculate the value of the following definite integral applying the Fundamental Theorem of Calculus:

Given the integral function $F(x) = \int_{0}^{x} \sqrt{t^2 + 9} \, dt$, use the Fundamental Theorem of Calculus to find the value of its derivative at the point $x = 4$, that is, calculate $F'(4)$.

A graphic designer is tracing the outline of a logo that follows the curve $f(x) = e^x$. Calculate the area enclosed under this curve, the $OX$-axis and the vertical lines $x = 0$ and $x = \ln(3)$.

Calculate the area of the plane region bounded by the parabola $f(x) = x^2 - 4x + 3$ and the x-axis ($OX$). Note: Remember to first find the points of intersection of the function with the axis.

Without calculating the antiderivative, justify using the properties of the definite integral why the value of the following integral is equal to $0$:

Find the area of the region bounded between the graphs of the functions $f(x) = 6 - x^2$ and $g(x) = 2$.

The speed of a remote-controlled car (in $m/s$) is given by the function $v(t) = 2t + 1$. Determine the total distance traveled by the car between the times $t = 1$ s and $t = 4$ s by calculating the definite integral of the speed.

An electronic components company has determined that the marginal cost of producing $x$ units of a sensor is $C'(x) = 3x^2 - 12x + 15$ dollars per unit. Calculate the total increase in cost when increasing production from $10$ to $20$ sensors.

Calculate the area of the bounded plane region limited by the graphs of the functions $f(x) = -x^2 + 5x$ and $g(x) = x + 2$. Graphically represent the region before performing the calculation.

Let the function $F(x)$ be defined by the following integral:

$F(x) = \int_{2}^{x^2-1} \frac{\ln(t)}{t} dt$

Find the equation of the tangent line to the graph of $F(x)$ at the point with x-coordinate $x = \sqrt{3}$.

A design studio is projecting the logo for a new technology company. The base of the logo is bounded by the function $f(x) = x^3 - 4x$ and the x-axis $OX$. Calculate the total area of the two closed regions that the curve forms with the axis.

Determine the value of the positive parameter $k > 0$ so that the area of the region bounded by the parabola $y = kx^2$ and the line $y = 4$ is exactly $16$ square units.

Find the area of the region bounded by the curve $f(x) = x^2 - 4x + 5$, the tangent line to this curve at the point $(3, f(3))$ and the y-axis $OY$.

Calculate the area of the region in the first quadrant enclosed by the graphs of $f(x) = \frac{8}{x}$, the line $y = x$ and the line $y = 4$.

An autonomous vehicle moves in a straight line and its speed (in ft/s) is given by the function $v(t) = t^2 - 6t + 8$, where $t$ is the time in seconds. Calculate the total distance traveled by the vehicle between the time $t = 0$ and $t = 5$. Note: Bear in mind that distance is the integral of the absolute value of the speed.

Consider the region bounded by the curves $y = e^x$, $y = e^{-x}$ and the vertical line $x = \ln(2)$.

1. a) Draw a sketch of the region.
2. b) Calculate the area of this region.

Let the function $f(x)$ be defined by the following expression involving an integral:

$F(x) = \int_{0}^{x^2} \frac{t-1}{2+e^t} dt$

You are asked to determine with reasoning:

1. a) The intervals of increase and decrease of $F(x)$ in its entire domain.
2. b) The x-coordinate of the points where $F(x)$ reaches its relative extrema, classifying them as maxima or minima.
3. c) Does the function $F(x)$ have any points of inflection in the interval $(0, +\infty)$? Justify your answer using the second derivative.

A team of engineers is designing the entrance arch to a technological tunnel. The shape of the upper arch is given by the function $f(x) = 4 - x^2$ for $x \in [-2, 2]$. To reinforce the structure, a straight line $y = k$ (where $0 < k < 4$) must be drawn in such a way that the area of the upper section (above the reinforcement line) is exactly twice the area of the lower section (between the reinforcement line and the ground, located on the $OX$-axis).

Calculate the exact value of the constant $k$ that satisfies this condition.

Consider the region of the plane $R$ bounded by the curve $y = \ln(x)$, the tangent line to this curve at the point with x-coordinate $x = e$, and the x-axis $OX$.

1. a) Find the equation of the mentioned tangent line.
2. b) Calculate the area of the region $R$.
3. c) Calculate the volume of the solid of revolution generated by rotating the region $R$ about the $OX$-axis.

A surveillance drone moves in a straight line. Its speed in feet per second is given by the absolute value function:

$v(t) = |t^2 - 6t + 5|$ for $t \in [0, 6]$ seconds.

1. a) Determine the total distance traveled by the drone during the first 6 seconds.
2. b) Calculate the value of the time $T \in (0, 6)$ such that the distance traveled in the interval $[0, T]$ is exactly half of the total distance calculated in the previous section.

A metal part is to be manufactured whose cross-section is bounded by the curves $y = \sqrt{x}$, $y = x - 2$ and the y-axis $OY$.

1. a) Graphically represent the region and determine its points of intersection.
2. b) Calculate the area of this section.
3. c) If the part is obtained by rotating this region about the y-axis $OY$, set up (without needing to calculate) the integral or sum of integrals that would provide the volume of the part.

Calculate the value of the following limit applying the Fundamental Theorem of Calculus and L'Hôpital's Rule:

$\lim_{x \to 0} \frac{x \cdot \int_{0}^{x} e^{-t^2} dt}{1 - \cos(x^2)}$

State in detail the steps followed and the necessary hypotheses for the application of each theorem.

Given the function $f(x) = \frac{x^2}{x^2 + 1}$, consider the region bounded by the graph of $f(x)$, its horizontal asymptote and the vertical lines $x = -1$ and $x = 1$.

1. a) Identify the equation of the horizontal asymptote of the function.
2. b) Calculate the area of this region.
3. c) Determine the volume of the solid generated by rotating this region about its own horizontal asymptote.

(Hint: Perform a substitution or translation to make calculating the volume easier).

A graphic design company creates a logo using the common area to the graphs of the functions $f(x) = \sin(x)$ and $g(x) = \cos(x)$ in the interval $x \in [0, \pi]$.

1. a) Determine the points of intersection of both functions in this interval.
2. b) The logo is divided into two regions by the x-axis $OX$. Calculate the area of the region of the logo that lies above the $OX$-axis.
3. c) Justify whether the total area of the logo (considering absolute values) is symmetrical with respect to any value of $x$ in the given interval.