<p><strong>Definite Integrals</strong></p>
<p><strong>Definition</strong> (Definite integral) The <em>definite integral</em> from \(a\) to \(b\) of the function \(\mathrm{f} \) is defined as :</p>
<p style="text-align: center">\[\displaystyle\int\nolimits_{a}^{b} \mathrm{f}(x) \, \mathrm{d}x = \big[ \mathrm{F}(x) \big]_{a}^{b} = \mathrm{F}(b) - \mathrm{F}(a)\]</p>
<p>where \(F(x) = \displaystyle\int \mathrm{f}(x) \, \mathrm{d}x\) and \(a\) and \(b\) are the (<em>lower</em> and <em>upper</em>) <em>limits</em> of the integration.</p>
<p>Note that:</p>
<ol class="num-list"><li><div class="list-item-paragraph">In a definite integral, there is an absence of the constant of integration.</div></li><li><div class="list-item-paragraph">In the IB syllabus, for the definite integral to be defined, we assume that the integrand \(\mathrm{f}\) is (piecewise) <em>continuous</em> on \the closed bounded interval \([a,b]\).</div><div class="list-item-paragraph">For example, <br>\[
\displaystyle\int\nolimits_{-1}^1 \frac{1}{x} \, \mathrm{d}x
\]<br>is undefined.</div></li><li><div class="list-item-paragraph">The Graphic Display Calculator (GDC) is very useful in evaluate definite integrals; this is especially the case when it is difficult, or even impossible, to obtain an antiderivative of the integrand.  You are warned that the radian mode be turned on whenever the integrand involves trigonometric functions.</div></li></ol>

**Definition** (Definite integral) The *definite integral* from \(a\) to \(b\) of the function \(\mathrm{f} \) is defined as :

\[\displaystyle \displaystyle\int\nolimits_{a}^{b} \mathrm{f}(x) \, \mathrm{d}x = \big[ \mathrm{F}(x) \big]_{a}^{b} = \mathrm{F}(b) - \mathrm{F}(a)\]

where \(F(x) = \displaystyle \displaystyle\int \mathrm{f}(x) \, \mathrm{d}x\) and \(a\) and \(b\) are the (*lower* and *upper*) *limits* of the integration.

In a definite integral, there is an absence of the constant of integration.

In the IB syllabus, for the definite integral to be defined, we assume that the integrand \(\mathrm{f}\) is (piecewise) *continuous* on \the closed bounded interval \([a,b]\).

For example, 
\[
\displaystyle \displaystyle\int\nolimits_{-1}^1 \frac{1}{x} \, \mathrm{d}x
\]
is undefined.

The Graphic Display Calculator (GDC) is very useful in evaluate definite integrals; this is especially the case when it is difficult, or even impossible, to obtain an antiderivative of the integrand.  You are warned that the radian mode be turned on whenever the integrand involves trigonometric functions.

Definition of definite integrals

Definition of definite integrals