Definite Integral

College/University

Definition

An integral that has start and end values. You can find the definite integral by finding the indefinite integral at the two points specified in the interval and then subtract them. You can then find the area that is between a functions' graph and the x-axis.

Worked examples

\(\int_{1}^{3} 2x\,dx = \left[x^2\right]_{1}^{3} = 3^2 - 1^2 = 8\)

Find the antiderivative, evaluate at the upper limit, then subtract the value at the lower limit.

\(\int_{0}^{\pi} \sin(x)\,dx = \left[-\cos(x)\right]_{0}^{\pi} = -\cos(\pi) - (-\cos(0)) = 2\)

The definite integral gives the net signed area between the curve and the x-axis on the interval.

Common mistakes

\(\int_{1}^{3} 2x\,dx = x^2\) → \(\int_{1}^{3} 2x\,dx = \left[x^2\right]_{1}^{3} = 8\) You must evaluate the antiderivative at both limits and subtract; don't leave it as a function.
\(\int_{3}^{1} 2x\,dx = 3^2 - 1^2 = 8\) → \(\int_{3}^{1} 2x\,dx = 1^2 - 3^2 = -8\) Always subtract (lower limit evaluation) from (upper limit evaluation); reversing limits flips the sign.
\(\int_{a}^{b} f(x)\,dx\) always equals the area under the curve → It gives net signed area; regions below the x-axis count as negative The definite integral sums signed area; use absolute value if you want total area regardless of sign.

Where you'll use it next

Definite integrals are essential for computing areas, volumes of revolution, arc lengths, work, and probability in calculus II and III, differential equations, physics, and statistics.

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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