Fundamental Theorem of Integral Calculus and 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 $\int_1^8 2 \times f(x) \, dx$.

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.

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 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.

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.

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 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.

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.

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 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 meters. Calculate the value of the definite integral on the interval $[0, 2]$ using Barrow's Rule.

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:

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

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 Barrow's Rule.

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

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.

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.

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.

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 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 $f(x) = x^2 - 4$, 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 Barrow's Rule to calculate the exact value of $H(3)$.

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.

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.

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.

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.