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 $\mathrm { c } _ { \mathrm { k } }$.
-$$f ( x ) = - 3 x ^ { 2 } , [ 0,4 ] , \text { midpoint }$$

Question

Quizplus · Accepted Answer

Since we are using the midpoint for each subinterval, we can divide the interval [0,4] into 4 equal subintervals of length 1.

The midpoint of the first subinterval [0,1] is x = 0.5. Therefore, we draw a rectangle with height f(0.5) = -3(0.5)^2 = -0.75 and width 1.

The midpoint of the second subinterval [1,2] is x = 1.5. Therefore, we draw a rectangle with height f(1.5) = -3(1.5)^2 = -6.75 and width 1.

Similarly, the midpoints of the third and fourth subintervals [2,3] and [3,4] are x = 2.5 and x = 3.5, respectively. Therefore, we draw rectangles with heights f(2.5) = -3(2.5)^2 = -18.75 and f(3.5) = -3(3.5)^2 = -38.25, both with width 1.

The area of these four rectangles gives us the Riemann sum: (-0.75)(1) + (-6.75)(1) + (-18.75)(1) + (-38.25)(1) = -64.5.

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 $\sum _ { k = 1 } ^ { 4 } f \left( c _ { k } \right) \Delta x _ { k }$