Exercises: Properties of the definite integral

Given two continuous functions $f(x)$ and $g(x)$ on the interval $[1, 5]$, we know the following values of their definite integrals:

Use the linearity properties of the integral to calculate the value of:

A logistics company monitors the speed of one of its vehicles. The graph of the function $f(x)$ represents the rate of fuel consumption. Knowing that:

Calculate, applying the interval additivity property, the value of the integral $\int_0^7 f(x) \, dx$ and interpret its meaning if the x-axis represents the time in hours.

Without calculating the antiderivative, reason based on the symmetry properties of functions why the value of the following definite integral is exactly zero:

If it is known that the integral of a function $f(x)$ over a reversed interval is $\int_8^3 f(x) \, dx = 14$, determine the value of the integral in the conventional direction:

Justify which property of the definite integral you have used.

Determine, without needing to calculate the antiderivative or evaluate the function, the value of the following integral:

State the name of the property on which you base your answer.

It is known that a function $f(x)$ is even and that the area enclosed under its curve between the y-axis ($x=0$) and the line $x=3$ is $4.5$ square units. That is:

Calculate the value of $\int_{-3}^3 f(x) \, dx$, justifying the result using the symmetry of the function.

Given a continuous function $h(x)$, the following data are known:

Calculate the value of the integral $\int_3^6 h(x) \, dx$ using the property of the sum of intervals.

A drone moves horizontally at a constant speed of $v(t) = 12$ metres per second. Calculate the distance travelled by the drone between $t=5$ and $t=15$ seconds by setting up the definite integral of the constant function:

Check the result using the geometric formula for the area of a rectangle. The figure shades the region between the graph of $v(t)$ and the time axis for $5 \le t \le 15$.

Cian is solving review exercises for the Leaving Certificate exam and comes across the following statement: "Knowing that $f(x)$ is a continuous function on $[1, 4]$ and that $\int_{1}^{4} f(x) \, dx = 8$, calculate the value of $\int_{1}^{4} [2 \times f(x) + 3] \, dx$".\nIn his notebook, he writes the following resolution:\n$$\int_{1}^{4} [2 \times f(x) + 3] \, dx = 2 \times \int_{1}^{4} f(x) \, dx + 3 = 2 \times 8 + 3 = 19$$\na) Identify with mathematical precision the mistake that Cian made. Explain why his treatment of the constant term $+3$ is incorrect considering the definition of the definite integral as the sum of areas of infinitesimal rectangles.\nb) Correct the calculation by rigorously applying the linearity properties of the definite integral.

Consider the following definite integral over a symmetric interval:\n$$I = \int_{-3}^{3} (4x^3 - 2x + 5) \, dx$$\na) Estimate: Before calculating antiderivatives, analyze the three terms of the integrand and their behavior with respect to the origin of coordinates. Predict in a reasoned manner what the exact numerical value of $I$ will be, justifying which terms will cancel out due to symmetry and which one will determine the final result.\nb) Check: Apply the linearity and symmetry properties (evenness and oddness) of the definite integral to break down $I$ into sums of simpler integrals. Formally justify the value of each one without calculating antiderivatives of the odd powers and check whether your initial estimation was correct.

It is known that an electrical signal received by a sensor follows a behaviour described by two functions, $f(x)$ and $g(x)$. After conducting several measurements in the laboratory, the following values were obtained for their definite integrals over the interval $[1, 4]$:

$\int_{1}^{4} f(x) \, dx = 7$ and $\int_{1}^{4} g(x) \, dx = -3$

Using the linearity properties of the definite integral, calculate the value of the integral resulting from the combination of both signals in that interval:

A surveillance drone travels along a straight path whose velocity as a function of time is given by a continuous function $v(t)$. Data analysts have determined that the accumulated displacement over different sections is:

1. From the start to second 3: $\int_{0}^{3} v(t) \, dt = 45$ metres.

2. From second 3 to second 8: $\int_{3}^{8} v(t) \, dt = 120$ metres.

3. From second 5 to second 8: $\int_{5}^{8} v(t) \, dt = 70$ metres.

Calculate, justifying the use of the interval additivity property, the displacement of the drone between seconds 0 and 5, that is: $\int_{0}^{5} v(t) \, dt$.

Consider a function $f(x)$ that is even and a function $g(x)$ that is odd across the set of real numbers. The following data are known:

$\int_{0}^{3} f(x) \, dx = 5$ and $\int_{-3}^{0} g(x) \, dx = 4$

Find, by applying the symmetry properties of the definite integral, the value of:

Let $f(x)$ be a continuous function on the interval $[2, 6]$ whose minimum value is known to be $m = 1$ and whose maximum value is $M = 5$.

1. a) State, giving reasons, between which values the integral $\int_{2}^{6} f(x) \, dx$ lies.

1. b) If we also know that $\int_{2}^{6} f(x) \, dx = 12$, find the mean value of the function on that interval and explain its geometric meaning.

Given the continuous function $f(x)$ defined on the interval $[-2, 5]$, it is known that the area enclosed by the curve and the $OX$ axis is as follows:

• On the interval $[-2, 1]$, the function is negative and the area of the region is $4$ $u^2$.

• On the interval $[1, 5]$, the function is positive and the area of the region is $10$ $u^2$.

Calculate, with reasoning, the values of the following definite integrals:

1. $\int_{-2}^{5} f(x) \, dx$

2. $\int_{5}^{1} f(x) \, dx$

3. $\int_{-2}^{5} |f(x)| \, dx$

Knowing that the integral of a function $f(x)$ on the interval $[a, b]$ is equal to $I$, prove using the properties of the definite integral that:

Explain the interpretation of this result in relation to the Mean Value Theorem for Definite Integrals.

Calculate the value of the definite integral of the function $f(x) = |x^2 - 4|$ on the interval $[0, 3]$. To do this, you must split the integral into several sections using the additivity property according to the sign of the integrand.

Without needing to calculate the antiderivatives, justify with reasoning which of the following integrals is greater using the integral comparison property ($f(x) \leq g(x) \implies \int f \leq \int g$):

Would your answer change if the interval of integration were $[1, 2]$? Justify your answer by analysing the behaviour of the functions on both intervals.

The Electricity Supply Board (ESB) monitors the net inflow rate $q(t)$ (measured in $\text{hm}^3/\text{day}$) in a reservoir in the Wicklow Mountains during a three-day rainfall event ($t \in [0, 3]$). The automatic sensors recorded the following integrated water measurements:\n- Total net inflow during the entire three days: $\int_{0}^{3} q(t) \, dt = 14.5 \, \text{hm}^3$.\n- Accumulated net inflow over the last two days: $\int_{1}^{3} q(t) \, dt = 9.2 \, \text{hm}^3$.\n- Control measurement between two floodgates recorded in reverse chronological order: $\int_{2}^{1} q(t) \, dt = -3.8 \, \text{hm}^3$.\na) Model the situation by setting up the relevant relationships using the properties of interval additivity and reversal of integration limits, and calculate the net volume delivered during the first day ($t \in [0, 1]$).\nb) Determine in a reasoned manner the net volume of water that entered the reservoir exclusively during the third day ($t \in [2, 3]$).

The aerodynamics department of a racing team models the cross-section of a wing using a continuous function $f(x)$ that is even on all of $\mathbb{R}$ ($f(-x) = f(x)$). In wind tunnel tests, the following values of accumulated lift have been determined:\n$$\int_{0}^{4} f(x) \, dx = 12 \qquad\text{and}\qquad \int_{-4}^{6} f(x) \, dx = 19$$\na) Using the symmetry properties of even functions and interval additivity, deduce with justification the value of $\int_{4}^{6} f(x) \, dx$ and that of $\int_{-6}^{-4} f(x) \, dx$.\nb) Inverse problem: It is desired to add a trim device that generates a uniform constant effect. Formulate and solve the equation that allows determining the exact value of the real constant $k$ so that the following equilibrium condition is met:\n$$\int_{-4}^{4} [k \times f(x) - 2] \, dx = 16$$

In digital audio signal processing, functions that represent complex sound waves are studied. Consider the function $f(x) = x^3 \times \cos(x^2) \times e^{\sin^2(x)}$. Without computing any antiderivative, justify your reasoning based on the properties of the definite integral to find the value of:

Then, if it is known that $\int_{0}^{5} f(x) \, dx = K$, determine in terms of $K$ the value of $\int_{-5}^{0} f(x) \, dx$.

Let the function $F(x)$ be defined for $x > 0$ by the following integral with variable limits:

Using the Fundamental Theorem of Calculus and the chain rule, calculate the value of the derivative $F'(1)$. Explain which property of the definite integral allows you to split the interval of integration to apply the theorem when the limits are not constant.

In an optics experiment on light diffraction, an integral of the form $I = \int_{0}^{\pi} \frac{x \times \sin(x)}{1 + \cos^2(x)} \, dx$ is obtained.

To solve it, use the reflection property $\int_{a}^{b} f(x) \, dx = \int_{a}^{b} f(a+b-x) \, dx$ to show that:

From this equality, find the exact value of the integral $I$.

Given two continuous functions $f$ and $g$ on the interval $[0, 1]$ such that $f(x) = \frac{1}{1+x^4}$ and $g(x) = 1 - x^4$, use the boundedness property of the definite integral to prove the following inequality without calculating the integrals:

Hint: Analyse the behaviour of the functions inside the interval $(0, 1)$ and use the fact that for $0 < u < 1$, $1-u < \frac{1}{1+u} < 1$ holds.

Let the function $f(x) = |x - a| + |x - 2|$ be defined on the interval $[0, 4]$. Given that the value of the definite integral of this function on that interval is a minimum for a specific value of $a \in [0, 4]$, use the properties of the integral to express $\int_{0}^{4} f(x) \, dx$ as a function of $a$, $I(a)$, and find the value of $a$ that minimises this integral.

A surveillance drone moves in a straight line with a velocity given by the function $v(t) = t \times \sqrt{16 - t^2}$ in metres per second, where $t$ is the time in seconds from the start of the mission up to $t = 4$.

Find all times $c \in (0, 4)$ at which the instantaneous velocity of the drone equals its average velocity over the entire interval $[0, 4]$. To do this, apply the Mean Value Theorem for Definite Integrals and evaluate the corresponding definite integral.

Given the integral function $G(x) = \int_{0}^{x} (t^2 - 3t + 2) \times e^{-t^2} \, dt$, solve the following parts using the Fundamental Theorem of Calculus:

1. a) Calculate $G'(x)$, study its sign and analyze the intervals where $G(x)$ is increasing and decreasing in $[0, 3]$.

1. b) Determine the abscissas of the relative maxima and minima of $G(x)$ in the interval $[0, 3]$.

1. c) Without calculating the integral, justify whether the value of $\int_{1}^{2} (t^2 - 3t + 2) \times e^{-t^2} \, dt$ is positive or negative, based on the sign of the integrand in that interval.

A mechanical part has a top profile defined by the function $f(x)$, which is periodic with period $T=2$. It is known that in the interval $[0, 2]$, the function is $f(x) = x(2-x)$.

Calculate the total area under the curve in the interval $[0, 20]$ using the properties of the definite integral for periodic functions and additivity over intervals.

In an advanced mathematics exam, students are asked to calculate the exact value of the definite integral:\n$$I = \int_{-2}^{2} \left( \frac{x^5 \times \sqrt{4 + x^2} + \sin^3(x)}{1 + \cos^2(x) + x^4} + 3x^2 + 5 \right) \, dx$$\nTwo students discuss the appropriate strategy:\n- Aoife (Method 1): Proposes first finding a general antiderivative $F(x)$ for the entire integrand using standard indefinite integration techniques (substitution and trigonometry) and then applying Barrow's rule by calculating $F(2) - F(-2)$.\n- David (Method 2): Proposes breaking down the integral by applying linearity, analytically studying the parity of the fraction, and solving the polynomial term by taking advantage of the symmetry properties of the definite integral.\na) Choose the method: Critically analyze the feasibility of both methods at 6th Year (Leaving Certificate) level. Justify rigorously why Aoife's approach is impracticable and why David's method is the appropriate one.\nb) Following the most efficient method, formally prove the symmetry of each term and calculate the exact numerical value of $I$, justifying each property used.

Seán and Aoife discuss the scope of the theoretical properties of the definite integral. Seán writes two conclusions on the board:\n1. Conclusion 1: «To calculate $\int_{2}^{8} f(x) \, dx$ knowing only that $\int_{2}^{10} f(x) \, dx = 15$ and that $\int_{8}^{10} f(x) \, dx = 6$, you cannot apply the additivity property $\int_a^c f + \int_c^b f = \int_a^b f$ because the point $c = 10$ is not inside the interval $[2, 8]$».\n2. Conclusion 2: «If for a continuous function $g(x)$ it is verified that $\int_{-3}^{3} g(x) \, dx = 0$, it is proven that $g(x)$ is necessarily an odd function on $[-3, 3]$».\na) Identify the conceptual error in Conclusion 1. Formally prove that the additivity property remains valid even if $c$ lies outside the interval $[a, b]$, combining the standard relation on $[2, 10]$ with the reversal of limits, and calculate the exact value of $\int_{2}^{8} f(x) \, dx$.\nb) Refute Conclusion 2 by explaining the error in logical reasoning (confusion between implication and converse) and provide an explicit counterexample: define a simple continuous function that is even on $[-3, 3]$ (and not identically zero) whose integral on that symmetric interval is exactly $0$.