Calculus I - Definition of the Definite Integral

Section 5.6 : Definition of the Definite Integral

8. For $ \displaystyle \int_{1}^{4}{{3x - 2\,dx}}$ sketch the graph of the integrand and use the area interpretation of the definite integral to determine the value of the integral.

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Here is the graph of the integrand, $f\left( x \right) = 3x - 2$, on the interval $\left[ {1,4} \right]$.

$The graph of the integrand $f\left( x \right) = 3x - 2$ on the interval $\left[ {1,4} \right]$. It is a straight line rising from $\left( {1,1} \right)$ to $\left( {4,10} \right)$ and the region between the line and the x-axis is shaded in.$

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Now, we know that the integral is simply the area between the line and the $x$-axis and so we should be able to use basic area formulas to help us determine the value of the integral. Here is a “modified” graph that will help with this.

$The graph of the integrand $f\left( x \right) = 3x - 2$ on $\left[ {1,4} \right]$ with the shaded area split up into two simpler pieces. A horizontal dashed line has been drawn from $\left( {1,1} \right)$ to $\left( {4,1} \right)$, which splits the shaded region into a rectangle of width 3 and height 1 along the bottom and a triangle of base 3 and height 9 sitting on top of it. The points $\left( {1,1} \right)$, $\left( {4,1} \right)$ and $\left( {4,10} \right)$ are marked and labeled.$

From this sketch we can see that we can think of this area as a rectangle with width 3 and height 1 and a triangle with base 3 and height 9. The value of the integral will then be the sum of the areas of the rectangle and the triangle.

Here is the value of the integral,

\[\int_{1}^{4}{{3x - 2\,dx}} = \left( 3 \right)\left( 1 \right) + \frac{1}{2}\left( 3 \right)\left( 9 \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{33}}{2}}}\]