Exercises: Fundamental theorem of calculus and Barrow's rule

A solar thermal panel has a design whose upper cross-section is bounded by the function $f(x) = 3x^2 + 2$, where $x$ is measured in metres. Calculate the value of the definite integral on the interval $[0, 2]$ using Barrow's Rule.

Given a function $f(x)$ continuous on the interval $[1, 8]$, the values of the following definite integrals are known:

• $\int_1^5 f(x) \, dx = 7$

• $\int_5^8 f(x) \, dx = -3$

Use the properties of the definite integral to calculate the value of $\int_1^8 2 \times f(x) \, dx$.

A video game programmer is designing the trajectory of a projectile. The velocity of the object as a function of time $t$ is given by the expression $v(t) = \int_1^t (3x^2 - 4x) \, dx$.

Apply the Fundamental Theorem of Calculus to find the expression for the instantaneous acceleration $a(t)$, knowing that acceleration is the derivative of velocity with respect to time ($a(t) = v'(t)$).

Calculate the area of the plane region bounded by the graph of the function $f(x) = \sin(x)$, the x-axis $OX$ and the vertical lines $x = 0$ and $x = \pi$. Sketch the region before applying the Fundamental Theorem of Calculus.

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

Find the value of the derivative of the following integral function at the point $x = 2$:

$$F(x) = \int_{0}^{x} \sqrt{t^3 + 1} \, dt$$

Explain which mathematical theorem you are using to solve the problem without needing to calculate the antiderivative.

The electricity consumption of a household during the first hour of the morning is modelled by the function $f(x) = e^x + x$, where $x$ is the time in hours. Calculate the total accumulated consumption, represented by the definite integral:

$$\int_{0}^{1} (e^x + x) \, dx$$

Without doing complex integration calculations, justify with reasoning why the value of the following integral is equal to $0$, based on the properties of odd functions and the symmetry of the interval:

$$\int_{-2}^{2} (x^5 + 3x^3 - x) \, dx$$

The figure shows the graph of $f(x)=x^5+3x^3-x$ on the interval $[-2,\,2]$, with the regions between the curve and the $X$-axis shaded.

Determine the area of the region enclosed between the curve $f(x) = x^2 - 4$ and the $OX$ axis on the interval $[0, 2]$. Remember that if the function is negative on the interval, the area corresponds to the modulus of the definite integral.

In a Leaving Certificate exam, students are asked to calculate the derivative of the function $F(x) = \int_2^{x^3} \sqrt{t^2 + 5}\,dt$ at the point $x = 1$. Aoife presents the following solution:

"By the Fundamental Theorem of Calculus, the derivative of an integral is the function inside evaluated at the upper limit, so $F'(x) = \sqrt{(x^3)^2 + 5} = \sqrt{x^6 + 5}$. Substituting $x = 1$, we get $F'(1) = \sqrt{1^6 + 5} = \sqrt{6}$".

1. a) Point out the conceptual error that Aoife makes and explain which rule or theorem she omitted applying.
2. b) State the version of the Fundamental Theorem of Calculus that must be applied when the upper limit of integration is a differentiable function $g(x)$.
3. c) Correct the exercise by calculating the exact and simplified value of $F'(1)$.

Consider the function $f(x) = 4 - x^2$ defined on the interval $[0, 2]$. We want to calculate the value of the definite integral $I = \int_0^2 (4 - x^2)\,dx$.

1. a) Before calculating any antiderivative, reasonably bound the value of the integral: justify geometrically why $I$ must be strictly less than the area of the rectangle with vertices $(0,0)$, $(2,0)$, $(2,4)$, and $(0,4)$ (area 8) and strictly greater than the area of the triangle with vertices $(0,0)$, $(2,0)$, and $(0,4)$ (area 4). From this, state a reasonable numerical estimate for the value of $I$.
2. b) Apply Barrow's Rule to calculate the exact value of the integral $I$, detailing the choice of antiderivative used.
3. c) Check whether the exact result obtained agrees with your previous estimate and explain why, in this case, the value of the integral coincides exactly with the geometric area enclosed by the graph of $f$ and the x-axis.

Given the function $f(x) = |x^2 - 2x|$, calculate the value of the definite integral on the interval $[0, 3]$ using Barrow's Rule:

$$\int_0^3 |x^2 - 2x| \, dx$$

Briefly explain why it is necessary to split the interval of integration to solve this exercise.

Calculate the derivative function $F'(x)$ of the following integral function using the Fundamental Theorem of Calculus and the chain rule:

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

State the domain of definition of the function $F(x)$ so that the integral is well-defined.

Determine the value of the positive real parameter $a$ knowing that the area bounded by the curve $f(x) = 3x^2$, the $OX$ axis and the vertical lines $x = 0$ and $x = a$ is exactly equal to 27 units of area. Use Barrow's Rule to do this.

Let the function $G(x)$ be defined by:

$$G(x) = \int_{0}^{x} (t^2 - 4)e^{t} \, dt$$

Find the equation of the tangent line to the graph of $G(x)$ at the point with x-coordinate $x = 0$.

Note: Remember that to find the y-coordinate of the point of tangency you must calculate $G(0)$ and for the gradient you must apply the Fundamental Theorem of Calculus.

The velocity of a prototype electric vehicle during the first 4 seconds of a test is given by the function $v(t) = t^2 - 3t + 4$ (in m/s).

1. a) Calculate the total distance travelled by the vehicle between times $t = 1$ s and $t = 3$ s applying Barrow's Rule.
2. b) Justify whether the vehicle stops at any moment during that interval.

Find the mean value of the function $f(x) = \sqrt{x+1}$ on the interval $[0, 3]$ and determine the point $c \in (0, 3)$ whose existence is guaranteed by the Mean Value Theorem for Definite Integrals. Interpret the result geometrically.

Given the integral function $H(x) = \int_{0}^{x} \frac{t-1}{t^2+4} \, dt$:

1. a) Find the intervals of increase and decrease of $H(x)$.
2. b) Determine the coordinates of the local extrema (maximums or minimums) of $H(x)$ and classify them.

Calculate the value of the definite integral of the following piecewise function on the interval $[0, 2]$:

$$f(x) = \begin{cases} e^{2x} & \text{if } 0 \le x \le 1 \\ \frac{e^2}{x} & \text{if } 1 < x \le 2 \end{cases}$$

Check beforehand whether the function is continuous on $x = 1$ to ensure that it is integrable.

Given the function $F(x) = \int_1^x (t^2 - 2t)\times e^t\,dt$, defined for all $x \in \mathbb{R}$, you are asked to study its intervals of increase and decrease, as well as classify its relative extrema.

To tackle the exercise, two students propose different approaches:

• Cian proposes first finding an explicit antiderivative of $(t^2 - 2t)e^t$ by applying the integration by parts method twice, using Barrow's Rule to obtain an algebraic expression for $F(x)$ without integrals, and then differentiating that resulting function.

• Aoife proposes applying the Fundamental Theorem of Calculus directly to obtain $F'(x)$ without computing any antiderivative beforehand.

1. a) Evaluate both approaches: are both mathematically valid? Justify which of them is noticeably more efficient in this context.
2. b) Following the most advantageous method, determine the derivative function $F'(x)$ and identify all its critical points.
3. c) Analyze the sign of $F'(x)$ to determine the intervals of increase and decrease of $F$, and reasonably classify each critical point as a relative maximum or minimum.

Met Éireann records the net precipitation rate supplying a reservoir during a 6-hour storm using the net flow function $c(t) = 6t - t^2$ (expressed in cubic hectometres per hour, $\text{hm}^3/\text{h}$), where $t \in [0, 6]$ is the time in hours elapsed since the beginning of the storm. The net volume accumulated in the reservoir is modeled as $V(t) = \int_0^t c(u)\,du$.

1. a) Interpret the physical and mathematical meaning of $V'(t)$ from the Fundamental Theorem of Calculus. At what instant does water enter the reservoir most rapidly?
2. b) From the sign of $V'(t)$, justify in which time interval the added volume of water $V(t)$ is increasing and at what instant it reaches its maximum value on the interval $[0, 6]$.
3. c) Use the Fundamental Theorem of Calculus to calculate the total volume of water (in $\text{hm}^3$) that the storm contributed to the reservoir after 6 hours.

Let the function $f(t) = \frac{\ln(t)}{t^2}$ be defined for $t > 0$. A new integral function $G(x)$ is defined as:

$$G(x) = \int_{x}^{x^2} f(t) \, dt$$

1. a) Applying the Fundamental Theorem of Calculus and the chain rule, obtain the expression for the derivative $G'(x)$.
2. b) Determine the intervals of increase and decrease of $G(x)$ in its domain.
3. c) Find the value of $x > 1$ for which the function $G(x)$ reaches a local maximum.

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

$$\lim_{x \to 0} \frac{\int_{0}^{x^2} \left( e^{t^2} - 1 \right) \, dt}{x^6}$$

Explain each step with reasoning, justifying the differentiation of the integral function.

A team of engineers is designing the trajectory of a racing drone. The vertical velocity of the drone in the time interval $[0, 4]$ seconds is given by the function $v(t) = \int_{0}^{t} \frac{1}{1 + z^4} \, dz$.

1. a) Without explicitly calculating the integral, justify why the drone is always rising or remaining at the same height.
2. b) Calculate the vertical acceleration of the drone at time $t = 1$ second.
3. c) At what instant is the vertical acceleration maximum? Justify your answer by studying the second derivative of the position.

Given the function $f(x) = x^2 - 4$, consider the integral function $H(x) = \int_{0}^{x} |f(t)| \, dt$.

1. a) Express $H(x)$ as a piecewise function for values of $x \in [0, 5]$.
2. b) Study the differentiability of $H(x)$ at the point $x = 2$. Is the Fundamental Theorem of Calculus directly applicable at that point?
3. c) Use the Fundamental Theorem of Calculus (evaluating the definite integral) to calculate the exact value of $H(3)$.

Let $f(x)$ be a continuous function on $\mathbb{R}$ such that $\int_{0}^{1} f(x) \, dx = 2$. Find the value of the constant $k$ so that the following equality holds, knowing that the area under the curve on the left has been shifted and scaled:

$$\int_{0}^{2} f\left(\frac{x}{2}\right) \, dx + \int_{1}^{\sqrt{2}} x \times f(x^2 - 1) \, dx = k$$

Use the substitution method (change of variables) in the definite integrals to relate them to the initial given value.

The logo of a new streetwear brand is bounded by the graphs of the functions $f(x) = \operatorname{sen}(\pi x)$ and $g(x) = ax$, with $a > 0$, in the first quadrant.

1. a) Determine the value of $a$ so that the line $y = ax$ passes through the local maximum of the function $f(x)$ on the interval $[0, 1]$.
2. b) For the value of $a$ found, calculate using Barrow's Rule the area of the region enclosed between both graphs.

Let the function be defined by the integral $F(x) = \int_{0}^{x} \frac{t - 1}{e^t + 1} \, dt$.

1. a) Find the critical points of $F(x)$ and classify them using the first derivative test.
2. b) Study the concavity of $F(x)$ and determine if it has any points of inflection.
3. c) Is there any value of $x$ for which $F(x)$ is negative? Reason your answer based on the sign of the integrand and the limits of integration.

Consider the integral function $L(x) = \int_{1}^{x} \frac{\sqrt{1 + t^3}}{t} \, dt$ defined for $x \geq 1$.

1. a) Show that $L(x)$ is a strictly increasing function in its domain.
2. b) It is known that the graph of $L(x)$ has a tangent line at the point with x-coordinate $x = 2$ whose equation is $y = mx + n$. Calculate the value of the gradient $m$.
3. c) Determine the rate of change of the gradient (second derivative) at that same point.

A continuous function $f: (0, +\infty) \to \mathbb{R}$ satisfies the following integral equation for every real number $x > 0$:

$$\int_1^{x^2} f(t)\,dt = 4x^3 - 4x$$

1. a) Applying the Fundamental Theorem of Calculus and the chain rule, differentiate both sides with respect to $x$ and obtain an algebraic relationship between $f(x^2)$ and the variable $x$.
2. b) From that relationship, deduce the explicit analytical expression of $f(x)$ for any $x > 0$ and calculate the value of $f(4)$ with justification.
3. c) Apply Barrow's Rule to your expression for $f$ to calculate $\int_1^4 f(t)\,dt$ and verify that it matches the value predicted by the original integral equation evaluated at $x = 2$.
4. d) Justify rigorously, relying on the sign of $f(t)$ or on the monotonicity of the integral function, whether there is any other value $c > 0$ with $c \neq 1$ such that $\int_1^c f(t)\,dt = 0$.

We want to calculate the following indeterminate limit:

$$L = \lim_{x \to 0} \frac{x - \int_0^x \frac{1}{1 + t^4}\,dt}{x^5}$$

Two students consider different resolution strategies:

• Darragh attempts to first compute the indefinite antiderivative $\int \frac{1}{1 + t^4}\,dt$ by decomposing the denominator into irreducible real quadratic factors, and then apply the Fundamental Theorem of Calculus.

• Aoife claims that it is not necessary to obtain an elementary antiderivative and that the limit can be computed directly by combining L'Hôpital's rule with the Fundamental Theorem of Calculus.

1. a) Explain why Darragh's method is ill-advised and infeasible within the time allotted for a written test, and justify rigorously whether Aoife's method is applicable by checking the hypotheses of the Fundamental Theorem of Calculus and L'Hôpital's rule.
2. b) Evaluate the limit $L$ using Aoife's proposal.
3. c) If we consider the function $F(x) = \int_0^x \frac{1}{1 + t^4}\,dt$, find the equation of the tangent line to the graph of $F$ at the point with abscissa $x = 0$. Using the value of the limit obtained in part b), determine with reasoning the relative position of the curve $y = F(x)$ with respect to that tangent line in a neighborhood of $x = 0$ (distinguishing between $x > 0$ and $x < 0$).