Exercises: Properties of integrals

A group of students is designing a logo for their graduation. The top part of the logo follows the curve of a function $f(x)$. We know that:

• $\int_{1}^{5} f(x) \, dx = 12$

• $\int_{5}^{8} f(x) \, dx = 7$

Calculate, applying the property of additivity over intervals, the value of the total area under the curve from $x=1$ to $x=8$.

Reason, without calculating any antiderivative, why the value of the following definite integral is exactly zero. To do this, observe the symmetry of the function with respect to the origin (an odd function) and the limits of integration:

In a study of the environmental impact of a factory, a function $f(x)$ that measures gas emissions has been obtained. It is known that the definite integral over an interval coincides with the area if the function is positive.

Suppose that $f(x)$ is continuous and does not change sign on the interval $[2, 6]$. If we know that $\int_{2}^{6} f(x) \, dx = -15$, what can we state about the position of the graph of the function $f(x)$ with respect to the $OX$ axis on that interval? Justify your answer based on the properties of the sign of the integral.

Let $f(x)$ be a continuous function that represents the velocity of a drone, in meters per second, as a function of time $x$, in seconds. Applying the properties of the definite integral, state the value of:

Justify your answer by mentioning the property corresponding to equal limits of integration.

To estimate the accumulated marginal profit of a transport company between several branches, Marcos needs to calculate the value of $\int_{-2}^{3} f(x)\,dx$. In the records, he only has two previously calculated integrals of the continuous function $f$: $\int_{-2}^{1} f(x)\,dx = 5$ and $\int_{3}^{1} f(x)\,dx = 4$. Marcos states the following in his report: «Since the values $-2$, $1$, and $3$ are involved, it is enough to directly add both values: $\int_{-2}^{3} f(x)\,dx = 5 + 4 = 9$».

1. a) Identify and explain accurately the conceptual error that Marcos made when applying the properties of the definite integral.
2. b) State the property of reversing the limits of integration (change of limits) and the additivity property with respect to the interval of integration that Marcos should have considered.
3. c) Correct the reasoning and calculate the true value of $\int_{-2}^{3} f(x)\,dx$.

You are asked to calculate the value of the definite integral $I = \int_{-3}^{3} (8x^5 - 6x^3 + 5)\,dx$. Two classmates suggest different approaches:

• Alba's method: Find an antiderivative $F(x) = \frac{4}{3}x^6 - \frac{3}{2}x^4 + 5x$, substitute the limits $x = 3$ and $x = -3$, and calculate the difference $F(3) - F(-3)$ using Barrow's rule.

• Bruno's method: Use the linearity property to split the integral into terms and apply the symmetry properties of even and odd functions on symmetric intervals of the form $[-a, a]$.

1. a) Work out the procedure that Bruno would follow, identifying which terms correspond to odd functions and what the result of integrating them on $[-3, 3]$ is without having to evaluate powers.
2. b) Compare the computational workload and the risk of making arithmetic errors in both methods. What is the exact final value of the integral $I$?

In designing a new logo for a skate clothing brand, an even function $f(x)$ is used to draw the upper part of a symmetry. It is known that the area bounded by this function and the $OX$-axis between the centre ($x=0$) and the right-hand endpoint ($x=4$) is 12 square units.

Knowing that $f(x)$ is an even function ($f(x) = f(-x)$), calculate the value of the following definite integral using its properties:

Note: Remember the behaviour of odd functions on symmetric intervals.

The water flow in a tank on an organic farm varies throughout the morning according to a continuous function $f(t)$. The measurement sensors have recorded the following accumulated flow data (in m³):

1. $\int_{2}^{8} f(t) dt = 24$

2. $\int_{5}^{8} f(t) dt = 9$

Calculate, using the properties of the limits of integration, the value of:

Consider the function $f(x)$ whose graph is continuous throughout the interval $[0, 10]$. If we know that the value of the definite integral over the entire interval is $\int_{0}^{10} f(x) dx = 25$, calculate the value of the parameter $k$ so that the following equality, based on the linearity properties, holds:

A market study for a new mobile application estimates that the rate of monthly net profit (in thousands of euros) is given by a continuous function $B(t)$. After analysing the data for the first year, it is observed that:

• The accumulated profit for the first six months is $\int_{0}^{6} B(t) dt = 40$.

• The accumulated profit for the second six months is $\int_{6}^{12} B(t) dt = 55$.

If the company's CEO decides to apply a tax of 10% on the rate of profit (which is equivalent to multiplying the function by $0,9$) and, in addition, there are constant maintenance costs of $1$ thousand euros per month, set up and calculate the integral that represents the actual final profit of the company at the end of the year:

In a study on electricity consumption in an agricultural region during a heat wave, technicians from the distribution company model the supplementary instantaneous demand using the continuous function $f(x) = \frac{12}{x+3}$ (in megawatts, MW), where $x \in [0, 9]$ represents the days elapsed since the start of the episode.

1. a) Estimate: Without calculating antiderivatives, justify that the function $f(x)$ is strictly decreasing on the interval $[0, 9]$, determine its lower bound $m$ and its upper bound $M$, and apply the boundedness property of definite integrals ($m(b-a) \le \int_a^b f(x)\,dx \le M(b-a)$) to bound the total accumulated consumption $I = \int_{0}^{9} f(x)\,dx$ between two numerical values.
2. b) Verify: Find the antiderivative of $f(x)$ and calculate the exact value of $I$ using Barrow's rule (express the result in terms of $\ln(2)$, taking into account that $\ln(4) = 2\ln(2)$). Using the approximation $\ln(2) \approx 0,693$, find the approximate consumption to two decimal places and check whether the result is consistent with the interval estimated in the previous section.

A photovoltaic plant with battery storage in Extremadura feeds or absorbs power from the electrical grid depending on the time of day. The net power exchanged over an 8-hour period is modeled by the continuous function $p(t) = 6 - 1,5t$ (in MW), where $t \in [0, 8]$ measures the hours elapsed since 10:00 ($t=0$). When $p(t) > 0$, the plant injects energy into the grid; when $p(t) < 0$, it absorbs energy from the grid. The total accumulated net energy transferred since the initial moment is defined as $E(x) = \int_{0}^{x} p(t)\,dt$ (in MWh).

1. a) Apply the Fundamental Theorem of Calculus to find the derivative $E'(x)$. Without calculating the integral beforehand, determine at what exact hour ($x$) the maximum net energy fed into the grid is reached, justifying your answer using the sign of $p(t)$.
2. b) Split the total integral $\int_{0}^{8} p(t)\,dt$ into two contiguous intervals according to the sign of $p(t)$ and calculate the value of each of the two partial integrals. Interpret why the overall result on $[0, 8]$ does not represent the total gross energy mobilized, but rather a net balance, using the concept of the integral as a «signed area».

Let the function $F(x)$ be defined by an integral of the following form:

Without calculating the integral, use the Fundamental Theorem of Calculus and the chain rule to determine:

1. a) The expression for the derivative $F'(x)$.
2. b) The value of $F'(1)$ and explain its physical meaning if $F(x)$ represented the accumulated profit of an investment after $x$ years.

Calculate the value of the definite integral in the interval $[-2, 2]$ of the function $f(x) = |x^2 - 1| + x \cdot e^{x^2}$. To do this, use the property of additivity over intervals to manage the modulus and the property of odd functions to simplify the calculation of the second term. Justify each step in detail based on the properties of the integral.

Given the graph of the continuous function $f(x)$ shown below, define the integral function $G(x) = \int_{-2}^{x} f(t) \, dt$ in the interval $[-2, 4]$.

Based on the properties of the integral function and observing the graph:

1. a) State the intervals where $G(x)$ is increasing and decreasing.
2. b) Determine for which value of $x$ the function $G(x)$ reaches its absolute maximum in the interval $[-2, 4]$.
3. c) What is the value of the derivative $G'(0)$?

Prove, using the comparison and monotonicity properties of the definite integral, that the value of the integral $I = \int_0^1 e^{x^2} \, dx$ is bounded between the values $1$ and $e$.

That is, justify without finding the antiderivative (which is non-elementary) why the following holds:

How would these bounds change if the interval of integration were $[0, 0,5]$? Justify your answer.

An express courier company analyzes the deviation of its operational delivery costs with respect to the optimal midday schedule using a marginal cost function $f(x) = a x^2 + b x + c$, where $x \in [-2, 2]$ represents the hours of advance or delay relative to 14:00 ($x=0$). From the accounting records, three facts are known:

1. The function has even symmetry on the interval $[-2, 2]$, that is, the cost due to time deviation depends solely on the magnitude of the time deviation, not on its sign ($f(-x) = f(x)$).

2. At the optimal moment ($x=0$), the marginal cost is $2$ euros/hour ($f(0) = 2$).

3. The total accumulated cost over the 4 hours of the time frame, calculated as $\int_{-2}^{2} f(x)\,dx$, is exactly $24$ euros.

1. a) Reasonably deduce the value of the coefficient $b$ from the parity condition and express $\int_{-2}^{2} f(x)\,dx$ as a multiple of the integral over the positive semi-axis $[0, 2]$.
2. b) Determine the values of the parameters $a$ and $c$, and write the final expression of the function $f(x)$.
3. c) The cost accumulation function is defined as $G(x) = \int_{0}^{x} f(t)\,dt$ for $x \in [-2, 2]$. Without calculating the explicit formula for $G(x)$, apply the Fundamental Theorem of Calculus to justify whether $G(x)$ is increasing or decreasing over the entire interval and analyze whether its graph has any inflection point.