Applications of the Definite Integral

Calculate the value of the following definite integral applying Barrow's Rule:

A graphic designer is drawing the outline of a logo that follows the curve $f(x) = e^x$. Calculate the area enclosed under that 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 find the intersection points of the function with the axis first.

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

A design studio is designing 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 enclosed regions formed by the curve and the axis.

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

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

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

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 $y = 4$.

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

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.

1. b) Calculate the area of that region.

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

A graphic design company creates a logo shaped like the region bounded by the graphs of the functions $f(x) = \sin(x)$ and $g(x) = \cos(x)$ and by the vertical lines $x = 0$ and $x = \pi$ (that is, on the interval $x \in [0, \pi]$). The figure shows both graphs and the shaded logo.

1. a) Determine the points where the two functions intersect in that interval.

1. b) The logo is divided into two regions by the x-axis $OX$. Calculate the area of the part of the logo that lies above the $OX$ axis.

1. c) Justify whether the whole logo (the region between the two graphs on $[0, \pi]$) is symmetric with respect to some vertical line $x = a$ with $a \in [0, \pi]$.

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

An electronics company has determined that the marginal cost of producing $x$ units of a sensor is $C'(x) = 60x^2 - 240x + 300$ pesos per unit. Calculate the total increase in cost when increasing production from $10$ to $20$ sensors.

An autonomous vehicle travels in a straight line and its speed (in m/s) is given by the function $v(t) = t^2 - 6t + 8$, where $t$ is the time in seconds. Calculate the total distance travelled by the vehicle between time $t = 0$ and $t = 5$. Note: Remember that the distance is the integral of the modulus of the velocity.

A surveillance drone moves in a straight line. Its velocity in meters 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 the total distance calculated in the previous part.

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 $F'(4)$.

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 $x = \sqrt{3}$.

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

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

Determine, giving reasons:

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

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

A team of engineers is designing the entrance archway to a technology 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 such that the area of the upper section (above the reinforcing line) is exactly twice the area of the lower section (between the reinforcing 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 aforementioned 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 metal piece 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) Sketch the region and determine its points of intersection.

1. b) Calculate the area of that cross section.

1. c) If the piece 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 give the volume of the piece.

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

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 around its horizontal asymptote.

(Hint: Perform a change of variable or translation to facilitate the calculation of the volume).

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

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

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