Questions: Find a formula for RN for f(x)=5x+7 on [3,8]. (Use symbolic notation and fractions where needed.)

Solution Steps

To find the formula for $R_{N}$ for $f(x)=5x+7$ on $[3,8]$, we can use the formula for the Riemann sum: $R_{N} = \sum_{i=1}^{N} f(x_{i}^{_}) \Delta x$ where $\Delta x = \frac{b-a}{N}$ and $x_{i}^{_}$ is the midpoint of the $i$th subinterval.

The definite integral of $f(x)=5x+7$ over the interval $[3,8]$ is given by: \[ \int_{3}^{8} (5x+7) \, dx \]

Integrating $5x+7$ with respect to $x$: \[ \int_{3}^{8} (5x+7) \, dx = \left[ \frac{5}{2}x^2 + 7x \right]_{3}^{8} \]

Substitute the upper and lower limits of integration into the antiderivative: \[ \left[ \frac{5}{2}(8)^2 + 7(8) \right] - \left[ \frac{5}{2}(3)^2 + 7(3) \right] = \frac{5}{2}(64) + 56 - \frac{5}{2}(9) - 21 \]

Final Answer