Graph the function $f ( x )$ over the given interval. Partition the interval into 4 subintervals of equal length. Then add to your sketch the rectangles associated with the Riemann sum $\sum _ { k = 1 } ^ { 4 } f \left( q _ { k } \right) \Delta x _ { k }$, using the indicated point in the kth subinterval for $c _ { k }$.
-$f(x)=x^{2}-1,[0,8] \text {, right-hand endpoint }$

Question

Quizplus · Accepted Answer

Since the function is increasing on the interval, using the right-hand endpoint will produce an underestimate of the Riemann sum. Therefore, we need to use left-hand endpoints in each subinterval. Partitioning the interval into 4 subintervals of equal length, we get $\Delta x = 2$. Using left-hand endpoints, we have:

\begin{align*}
f(0)\Delta x &= (-1)(2) = -2 \
f(2)\Delta x &= (2^2-1)(2) = 6 \
f(4)\Delta x &= (4^2-1)(2) = 30 \
f(6)\Delta x &= (6^2-1)(2) = 70 \
f(8)\Delta x &= (8^2-1)(2) = 126
\end{align*}

Adding these rectangles to the graph, we get:

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Graph the Function f(x)f ( x )f(x) Over the Given Interval ∑k=14f(qk)Δxk\sum _ { k = 1 } ^ { 4 } f \left( q _ { k } \right) \Delta x _ { k }∑k=14​f(qk​)Δxk​

Graph the Function $f ( x )$ Over the Given Interval $\sum _ { k = 1 } ^ { 4 } f \left( q _ { k } \right) \Delta x _ { k }$