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 $\sum _ { k = 1 } ^ { 4 } f \left( c _ { k } \right) \Delta x _ { k }$, using the indicated point in the kth subinterval for $\mathrm { c } _ { \mathrm { k } }$.
-$$f ( x ) = x ^ { 2 } - 2 , [ 0,8 ] , \text { left-hand endpoint }$$

Question

Quizplus · Accepted Answer

To approximate the area, we need to draw rectangles under the curve with equal width of $2$ (since we have $4$ sub-intervals of equal length), starting from $x=0$ and ending at $x=8.$

The left-hand endpoint means that we need to evaluate the function at the left endpoint of each sub-interval and then multiply by the width of the rectangle. So, for the first rectangle, we need to evaluate $f(0)$ and multiply by $2$ to get the area of the first rectangle. Similarly, we find the area of the other rectangles and add them up to get the total area.

As we can see, the heights of the rectangles are below the curve. Therefore, this is an underestimate of the area. We can see that the approximation will be the most accurate if we choose the sample points to be as far away as possible from the curve. In this case, the left endpoints produce the biggest discrepancy between the Riemann sum and the actual area, leading to our underestimate of the area.

Out of the given choices, option B is the only one that depicts rectangles with left-hand endpoints and equal sub-intervals of width $2.$

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

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