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 meters. 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 Barrow's Rule.

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.

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.

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 Barrow's Rule 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.