Determine the integral $\int_0^bx^2\,dx$ using upper and lower sums.

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

Determine the integral $\int_0^bx^2\,dx$ using upper and lower sums.

Answer: 

$\int_0^b x^2\,dx=\frac{b^3}{3}$

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.

First, we know that if <Sign in to see all the formulas> then <Sign in to see all the formulas>. Therefore, we know that in each closed interval <Sign in to see all the formulas> that <Sign in to see all the formulas> will take on a minimum value of <Sign in to see all the formulas> and a maximum value of $x^2_i$. This gives us

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and

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Now for the "trick" to get this to work. We know from this problem that if <Sign in to see all the formulas> then

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Let's multiply <Sign in to see all the formulas> by <Sign in to see all the formulas> to get

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Finally, we see that

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Now we know that

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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 $[0,b]$. According to our definition of a definite integral, since the number <Sign in to see all the formulas> holds for all partitions of $[0,b]$ it is <Sign in to see all the formulas>. Recall that $x_0=0$ and $x_N=b$ so that

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