Find four different integrals that are represented by the following Riemann sum? \( \sum_{i=1}^{n}\left(\frac{3}{n}\right)\left[4-\frac{9 t^{2}}{n^{2}}\right] \) Intergrate the following:

To find four different integrals represented by the given Riemann sum \[ \sum_{i=1}^{n}\left(\frac{3}{n}\right)\left[4-\frac{9 t^{2}}{n^{2}}\right], \] we first need to analyze the components of the sum. 1. **Identify the width of the subintervals**: The term \(\frac{3}{n}\) suggests that the width of each subinterval is \(\Delta t = \frac{3}{n}\). 2. **Identify the function being summed**: The expression inside the brackets, \(4 - \frac{9 t^2}{n^2}\), can be interpreted as a function evaluated at some point \(t_i\) in the \(i\)-th subinterval. 3. **Determine the interval of integration**: Since \(\Delta t = \frac{3}{n}\), we can infer that the interval of integration is likely from \(0\) to \(3\) (as \(n\) approaches infinity, the sum approximates the integral over this interval). 4. **Express \(t_i\)**: We can express \(t_i\) as \(t_i = \frac{3i}{n}\), which is a typical choice for Riemann sums. Now, substituting \(t_i\) into the function gives: \[ 4 - \frac{9}{n^2} \left(\frac{3i}{n}\right)^2 = 4 - \frac{9 \cdot 9 i^2}{n^4} = 4 - \frac{81 i^2}{n^4}. \] Thus, the Riemann sum can be rewritten as: \[ \sum_{i=1}^{n} \left(\frac{3}{n}\right) \left[4 - \frac{81 i^2}{n^4}\right]. \] As \(n\) approaches infinity, this sum approaches the integral: \[ \int_{0}^{3} \left(4 - \frac{81}{9} t^2\right) dt = \int_{0}^{3} \left(4 - 9t^2\right) dt. \] Now, we can find four different integrals by varying the function inside the sum while keeping the same structure. Here are four possible integrals: 1. **Integral of the original function**: \[ \int_{0}^{3} \left(4 - 9t^2\right) dt. \] 2. **Integral with a constant added**: \[ \int_{0}^{3} \left(5 - 9t^2\right) dt. \] 3. **Integral with a different coefficient for \(t^2\)**: \[ \int_{0}^{3} \left(4 - 4t^2\right) dt. \] 4. **Integral with a different constant and coefficient for \(t^2\)**: \[ \int_{0}^{3} \left(2 - 9t^2\right) dt. \] These integrals represent variations of the original function while maintaining the same structure of the Riemann sum. Each integral can be computed to find the area under the respective curves over the interval \([0, 3]\).