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      <text>10.3.5 (Definitie integral definition)</text>
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      <text><![CDATA[<p>With the examination we have just completed, we can state the following definition of the definite integral:</p>
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<p>The <b>definite integral</b> \(\displaystyle\int_a^b f(x)\,dx\) is defined to be the limit of the Riemann sums of \(f\) on the interval \(a\leq x\leq b\), if this limit exists.</p><p>For positive functions, the value of the definite integral is interpreted as the area under the curve.</p>
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<p>So when we say \(\displaystyle\int_0^6 x^2\,dx=72\), we are really saying that if we kept increasing \(n\) on the slider above, both the left and right sums would eventually be as close to 72 as we like. And that is how we define the area under the curve \(y=x^2\) between 0 and 6.</p>
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<p>The main issue with this definition is that infinite sums are difficult to compute and work with (as we have seen in week 8). Fortunately, there is an easier way to compute integrals rather than working with this definition explicitly. We will introduce this in the next section.</p>]]></text>
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