Determine the integral $\int_a^bx\,dx$ using upper and lower sums.

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Lesson Parent: 
Integration III: Definite Integrals As Upper and Lower Sums
Problem: 

Determine the integral $\int_a^bx\,dx$ using upper and lower sums.

Answer: 

$\int_a^b x\,dx=\frac{b^2}{2}-\frac{a^2}{2}$

SAMPLE SOLUTION

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Solution: 

First, let's find $L(P)$ and $U(P)$ for any arbitrary partition <Sign in to see all the formulas>. (Keep in mind our notation that $a=x_0$ and $b=x_N$.) Recall that $L(P)$ uses the minimum value of $f(x)$ in each closed interval defined by $P$. Similarly, $U(P)$ will use the maximum value of $f(x)$ in each closed interval.

Since this is a straight line we immediately know that for each interval <Sign in to see all the formulas> that the minimum value will be <Sign in to see all the formulas> and the maximum value will be $x_i$. This gives us

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and

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Now here is the "trick" to get this to work. First, note that

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In other words, we know that the average of <Sign in to see all the formulas> and $x_i$ is between the two of them. Second, note that

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Finally, we have

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Now we know that

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Now in

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we have

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so that

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We can now see that

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Since we did this with an arbitrary partition, we know that <Sign in to see all the formulas> holds for all partitions of $[a,b]$. According to our definition of a definite integral, since the number <Sign in to see all the formulas> holds for all partitions of $[a,b]$ it is <Sign in to see all the formulas>. Recall that $x_0=a$ and $x_N=b$ so that

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