The graph $of$ a runction $f(x)is$ given below. $(Each$ grid square $is1$ unit wide and $1$ unit high.$)$

Use rectangles $to$ estimate the area under the curve $on$ the interval. $(Determine$ the number $of$ rectangles, the width $of$ the rectangles $in$ each case, and which sample points should

$be$ used $in$ your calculation using the given directions. When determining the height $of$ a rectangle, assume that the height $is$ a multiple $of0\mathrm{.}5.)$

$(a)On$ the interval $0,12$ using $4$ subintervals $of$ equal width and right endpoints $as$ sample points.

square units

$(b)On$ the interval $0,12$ using $12$ subintervals $of$ equal width and right endpoints $as$ sample points.

square units

$(c)On$ the interval $0,12$ using $6$ subintervals $of$ equal width and midpoints $as$ sample points.

square units

$A$~~${}_k^n$

Recall that the approximate area $is$ given $by{?}_{(}()1)S_k^{*},,$ where $S_k^{*}=f(x_k^{*})*x$ for some sample points $x_k^{*}in$ the subinterval $x_k-1,x_k.$ Did you find the partition determined $byn$

subintervals $of$ equal width $x$ and calculate the correct values for each $x_k^{*}?$ Did you determine the function values $f(x_k^{*})to$ the nearest $\frac{1}{2}?$