Find a formula for the Riemann sum obtained by dividing the interval [a, b] into n equal subintervals and using the right-hand endpoint for each $\mathrm { c } _ { \mathrm { k } }$. Then take a limit of these sums as $\mathrm { n } \rightarrow \infty$ to calculate the area under the curve over [a, b].
-$f ( x ) = 3 x + 6$ over the interval $[ 0,3 ]$.
A) $18 + \frac { 27 n ^ { 2 } + 27 n } { 2 n ^ { 2 } }$; Area $= \frac { 63 } { 2 }$
B) $18 + \frac { 24 n ^ { 2 } + 28 n } { 2 n ^ { 2 } } ;$ Area $= 30$
C) $18 + \frac { 27 n ^ { 2 } + 27 n } { 2 n ^ { 2 } }$; Area $= \frac { 63 } { 5 }$
D) $18 - \frac { 27 n ^ { 2 } + 27 n } { 2 n ^ { 2 } }$; Area $= \frac { 9 } { 2 }$

Question

Quizplus · Accepted Answer

The correct Riemann sum formula for $f(x) = 3x + 6$ using right endpoints is obtained by dividing the interval $[0, 3]$ into $n$ equal parts, each of length $\Delta x = \frac{b-a}{n} = \frac{3-0}{n} = \frac{3}{n}$. The right endpoint for the $k$th interval is $x_k = a + k\Delta x = 0 + k\frac{3}{n} = \frac{3k}{n}$. Substituting $x_k$ into $f(x)$ gives $f(x_k) = 3\left(\frac{3k}{n}\right) + 6$. The Riemann sum is then $\sum_{k=1}^{n} f(x_k)\Delta x = \sum_{k=1}^{n} \left(3\left(\frac{3k}{n}\right) + 6\right)\left(\frac{3}{n}\right)$, which simplifies to $18 + \frac{27n^2 + 27n}{2n^2}$ when evaluated. Taking the limit as $n \rightarrow \infty$ gives the area under the curve, which is $\frac{63}{2}$ for this linear function over the interval $[0, 3]$.

Find a Formula for the Riemann Sum Obtained by Dividing ck\mathrm { c } _ { \mathrm { k } }ck​

Find a Formula for the Riemann Sum Obtained by Dividing $\mathrm { c } _ { \mathrm { k } }$