Exercises: Applications of the definite integral

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

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 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 $0$:

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

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 $C'(x) = 3x^2 - 12x + 15$ euros per unit. Calculate the total increase in cost when increasing production from $10$ to $20$ sensors.

To calculate the area of the closed region bounded by the graph of the function $f(x) = x^3 - 4x$ and the x-axis on the interval $[-2, 2]$, Seán wrote the following on his exam:

He therefore concludes that the enclosed area is zero.

1. a) Explain in a reasoned way the conceptual error committed by Seán. Why does the definite integral equal zero even though the region has a non-zero geometric surface area?
2. b) Correct his solution: determine the x-intercepts of the function, set up correctly the integrals that allow calculating the actual area, and find its exact value by applying the Fundamental Theorem of Calculus.

You wish to calculate the area of the bounded region delimited by the parabola $f(x) = 3x - x^2$ and the x-axis ($y = 0$).

1. a) Before calculating any integral, make a reasoned estimate of the area by bounding it between two numerical values:

• Obtain an upper bound by calculating the area of a rectangle that circumscribes the region exactly (its base is the segment between the x-intercepts and its height is the y-coordinate of the parabola's vertex).

• Obtain a lower bound by calculating the area of the inscribed isosceles triangle whose base is that segment on the x-axis and whose upper vertex is the vertex of the parabola.

1. b) Now calculate the exact value of the area by applying Barrow's rule.
2. c) Check whether the exact result obtained falls within the bounded interval from part a) and justify whether your estimate was reasonable.

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.

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 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.

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.

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 $OY$.

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 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.

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.

During a rainstorm recorded by Met Éireann in the River Barrow basin in south-east Ireland, the inflow rate of water into a reservoir during the first $8$ hours of the event ($0 \le t \le 8$) is modeled by the function:

where $t$ is measured in hours and $c(t)$ in thousands of cubic meters per hour ($10^3\text{ m}^3/\text{h}$). Simultaneously, the dam maintains a constant discharge outflow rate of $22$ thousand cubic meters per hour to prevent overflowing.

1. a) Write the function $r(t) = c(t) - 22$, which represents the net rate of change of the volume of the reservoir, and determine in a reasoned way at what time $t \in [0, 8]$ the volume of stored water reaches its relative maximum during the storm. Justify your answer using the derivative of the accumulated volume function.
2. b) Knowing that at the start of the storm ($t = 0$) the reservoir had a reserve of $500$ thousand cubic meters of water, calculate using a definite integral the exact total volume stored at the moment of maximum accumulation found in the previous section.

Consider the parabola $f(x) = 4x - x^2$ and the family of lines passing through the origin of coordinates, with equation $y = mx$, where $m$ is a real parameter such that $0 < m < 4$.

1. a) Determine, in terms of $m$, the intersection points between the line and the parabola, and reason which of the two graphs lies above in the open interval between these points.
2. b) Express using a definite integral the area of the region enclosed between both curves and prove, applying Barrow's rule, that this area corresponds to the expression:

1. c) Formulate and solve the inverse problem: determine in a reasoned way what exact value the slope $m$ must take so that the area of the enclosed region is exactly equal to $4.5\text{ u}^2$.

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.

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 surveillance drone moves in a straight line. Its velocity in metres per second is given by the modulus function:

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

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

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.

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.

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).

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 $a \in [0, \pi]$.

You wish to calculate the area of the bounded planar region in the first quadrant delimited jointly by the curve $y = x^2$, the line $y = 2 - x$, and the x-axis ($y = 0$).

1. a) Design and describe two different analytical strategies to calculate this area:

• Method 1 (integration with respect to the variable $x$): determine the intersection points and necessary intervals, expressing the area as the sum of integrals with respect to $x$.

• Method 2 (integration with respect to the variable $y$): express the variable $x$ in terms of $y$ in each equation and write the total area as a single definite integral with respect to the variable $y$.

1. b) Compare both methods and justify in a reasoned way which one is more advantageous in terms of computational efficiency and a lower likelihood of making algebraic errors.
2. c) Apply the method you consider optimal (or combine integration with elementary Euclidean geometry on the linear segment) to obtain the exact numerical value of the area of the region.

In a hydraulic engineering project to modernize an irrigation network in the Shannon basin, a canal is designed whose cross-section has a parabolic profile symmetric with respect to the vertical axis, modeled by the function $y = ax^2$ (with $a > 0$, where $x$ and $y$ are measured in meters). The maximum water elevation is set by a horizontal line at a depth $y = h$ (with $h > 0$). The design must simultaneously satisfy two technical requirements:

1. The width of the upper water surface (distance between the two banks at height $y = h$) must be exactly $6\text{ m}$.

2. The cross-sectional wetted area (the surface enclosed between the parabola and the free surface $y = h$) must be $12\text{ m}^2$ to ensure the design flow rate.

1. a) From the first technical requirement, obtain a simple mathematical relationship that expresses the depth $h$ in terms of the curvature parameter $a$.
2. b) Set up the definite integral that allows calculating the wetted area and obtain its value in terms of $a$ and $h$ using Barrow's rule.
3. c) Determine the exact numerical values of the depth $h$ and the parameter $a$ that fulfill both project specifications.
4. d) Design reasoning: If due to an extraordinary increase in flow rate the water height doubles reaching a level $H = 2h$, reason analytically (relying on the properties of the integral and without needing to recalculate it numerically from scratch) whether the wetted area will double, be greater than double, or be less than double.