Properties of the definite integral
When you know the value of some integrals, you can often work out others without finding any antiderivative: you just need to apply a few properties. They are the same ones as always (linearity, splitting into pieces...) but now applied to specific numbers, not to functions.
Linearity: sums and multiples
The definite integral behaves just like the indefinite integral with respect to sums and constant multipliers.
Given two continuous functions $f(x)$ and $g(x)$ on the interval $[1, 5]$, we know the following values of their definite integrals: $$\int_1^5 f(x) \, dx = 6 \quad…
Given two continuous functions $f(x)$ and $g(x)$ on the interval $[1, 5]$, we know the following values of their definite integrals:
$$\int_1^5 f(x) \, dx = 6 \quad \text{and} \quad \int_1^5 g(x) \, dx = -3$$Use the linearity properties of the integral to calculate the value of:
$$\int_1^5 [2f(x) + 4g(x)] \, dx$$Additivity over the interval
If you split $[a,b]$ into two pieces with an intermediate point $c$, the integral over the whole interval is the sum of the integrals over the pieces. This is what you already used in the previous section to "cut up" a region where the sign changes.
A logistics company monitors the speed of one of its vehicles. The graph of the function $f(x)$ represents the rate of fuel consumption. Knowing that: $$\int_0^4 f(x) \,…
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:
$$\int_0^4 f(x) \, dx = 12 \quad \text{and} \quad \int_4^7 f(x) \, dx = 5$$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.
Given a continuous function $h(x)$, the following data are known: $$\int_1^3 h(x) \, dx = 4 \quad \text{and} \quad \int_1^6 h(x) \, dx = 10$$ Calculate the value of the…
Given a continuous function $h(x)$, the following data are known:
$$\int_1^3 h(x) \, dx = 4 \quad \text{and} \quad \int_1^6 h(x) \, dx = 10$$Calculate the value of the integral $\int_3^6 h(x) \, dx$ using the property of the sum of intervals.
Swapped limits and the point as an interval
Two special cases are worth knowing by heart because they appear constantly when manipulating limits of integration.
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…
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:
$$\int_3^8 f(x) \, dx$$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: $$\int_{\sqrt{2}}^{\sqrt{2}} \frac{e^x \cdot…
Determine, without needing to calculate the antiderivative or evaluate the function, the value of the following integral:
$$\int_{\sqrt{2}}^{\sqrt{2}} \frac{e^x \cdot \ln(x^2+1)}{\cos^2(x)+5} \, dx$$State the name of the property on which you base your answer.
Symmetry: even and odd functions
On a symmetric interval $[-a,a]$, knowing whether $f$ is even or odd saves you half the work (or all of it, if it is odd).
Without calculating the antiderivative, reason based on the symmetry properties of functions why the value of the following definite integral is exactly zero:…
Without calculating the antiderivative, reason based on the symmetry properties of functions why the value of the following definite integral is exactly zero:
$$\int_{-10}^{10} (x^7 - \sin(x) + x^3) \, dx$$It is known that a function $f(x)$ is even and that the area enclosed under its curve between the y-axis ($x=0$) and the line $x=3$ is $4,5$ square units. That is:…
It is known that a function $f(x)$ is even and that the area enclosed under its curve between the y-axis ($x=0$) and the line $x=3$ is $4,5$ square units. That is:
$$\int_0^3 f(x) \, dx = 4,5$$Calculate the value of $\int_{-3}^3 f(x) \, dx$, justifying the result using the symmetry of the function.
Consider a function $f(x)$ that is even and a function $g(x)$ that is odd across the set of real numbers. The following data are known: $\int_{0}^{3} f(x) \, dx = 5$ and…
Consider a function $f(x)$ that is even and a function $g(x)$ that is odd across the set of real numbers. The following data are known:
$\int_{0}^{3} f(x) \, dx = 5$ and $\int_{-3}^{0} g(x) \, dx = 4$
Find, by applying the symmetry properties of the definite integral, the value of:
$$\int_{-3}^{3} [3f(x) - g(x)] \, dx$$