Calculus I - Definition of the Definite Integral

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Section 5.6 : Definition of the Definite Integral

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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 \(\left[ {1,4} \right]\).

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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.

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,

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