Definite Integral Rules

College/University

Definition

Defnite integrals are integrals that have start and end values. The definite integral rules tell you how to treat definite integrals in different situations. Keep in mind that these rules only apply when the integrals exist.

Worked examples

\(\int_{a}^{b} [f(x) + g(x)]\,dx = \int_{a}^{b} f(x)\,dx + \int_{a}^{b} g(x)\,dx\)

You can split the integral of a sum into the sum of two separate integrals over the same interval.

\(\int_{a}^{b} c\cdot f(x)\,dx = c\int_{a}^{b} f(x)\,dx\)

A constant multiplier can be pulled out in front of the integral.

\(\int_{a}^{b} f(x)\,dx = -\int_{b}^{a} f(x)\,dx\)

Reversing the limits of integration flips the sign of the definite integral.

Common mistakes

\(\int_{a}^{b} f(x)\cdot g(x)\,dx = \int_{a}^{b} f(x)\,dx \cdot \int_{a}^{b} g(x)\,dx\) → \(\int_{a}^{b} f(x)\cdot g(x)\,dx\) cannot be split You can split sums but not products—there is no product rule for definite integrals.
\(\int_{0}^{2} f(x)\,dx + \int_{0}^{3} f(x)\,dx = \int_{0}^{5} f(x)\,dx\) → \(\int_{0}^{2} f(x)\,dx + \int_{2}^{3} f(x)\,dx = \int_{0}^{3} f(x)\,dx\) The additivity rule requires the upper limit of the first to match the lower limit of the second.
\(\int_{3}^{3} f(x)\,dx = f(3)\) → \(\int_{3}^{3} f(x)\,dx = 0\) When the limits are identical, the integral is zero—there is no interval to integrate over.

Where you'll use it next

You'll use definite integral rules to evaluate areas, volumes, and accumulated quantities in Calculus II, to simplify complex integrals in differential equations, and in physics for work and probability calculations.

Found in 1 StudyPug lesson

Definite integral

Calculus 2

In this section, we will evaluate definite integrals by calculating the area under the curve. We see that the region of integration depends on the lower limit and upper limit of the integral. These areas will be fairly easy to calculate since most of the areas under the curve involve shapes that are familiar to us. We will also notice that curves under the x-axis gives us negative area. Next, we will take a look at questions which involves sketching the curve ourselves, and then determining the area. Lastly, we will take a look at a unique question which involves finding the area of a specific region when given information about two definite integrals.

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