Properties of integrals

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.

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 $x=8$.

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:

$$\int_{5}^{2} f(t) dt$$

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:

$$\int_{0}^{12} [0,9 \cdot B(t) - 1] dt$$

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:

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

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:

$$\int_{-4}^{4} [f(x) + \frac{1}{2}x^3] dx$$

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

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:

$$1 \leq \int_0^1 e^{x^2} \, dx \leq e$$

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

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

$$F(x) = \int_{x^2}^{x^3} \sqrt{t^2 + 9} \, dt$$

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.

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 $G'(0)$?