Question

compute l₃ and r₃ for graphs a and b.
the value of l₃ for graph a is 13
(simplify your answer.)
the value of r₃ for graph a is 40
(simplify your answer.)

Explanation:

Step1: Calculate \( \Delta x \)

For \( n = 3 \), if the interval is from \( x = 0 \) to \( x = 4 \), then \( \Delta x=\frac{4 - 0}{3}=\frac{4}{3} \). But since we are counting unit - squares (assuming each square has side - length \( 1 \)), for Graph B:
The left - hand Riemann sum \( L_{3}\) formula is \( L_{3}=\sum_{i = 0}^{2}g(x_{i})\Delta x\), and the right - hand Riemann sum \( R_{3}\) formula is \( R_{3}=\sum_{i = 1}^{3}g(x_{i})\Delta x\). Here \( \Delta x = 2\) (because the interval from \( x = 0\) to \( x = 4\) is divided into \( n = 3\) sub - intervals, but by counting the grid, we can also use the height of the rectangles at the left and right endpoints).
For \( L_{3}\) (using the left - endpoints of the sub - intervals \([0,2]\), \([2,4]\)):
The height of the rectangle over \([0,2]\) is \(g(0)=8\), over \([2,4]\) (first two sub - intervals, since \(n = 3\) and we take left - endpoints) is \(g(2) = 4\).
\(L_{3}=(8\times2)+(4\times2)+(1\times2)\)

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Step2: Calculate \( R_{3}\)

For \( R_{3}\) (using the right - endpoints of the sub - intervals \([0,2]\), \([2,4]\)):
The height of the rectangle over \([0,2]\) is \(g(2)=4\), over \([2,4]\) is \(g(4)=1\).
\(R_{3}=(4\times2)+(1\times2)+(0\times2)\)

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Answer:

The value of \(L_{3}\) for Graph B is \(26\).
The value of \(R_{3}\) for Graph B is \(10\).