Let \( f(x)=\cos (3 x) \) on the interval \( [-4,8] \). Partition the interval \( [-4,8] \) into 4 equal subintervals. Find the left and right Riemann sums, \( L_{4} \) and \( R_{4} \), respectively. You only need to type in the expression that represents the sum without calculating it, for example, \( \cos (-1) \cdot 2+\cos (1) \cdot 2+\cos (3) \cdot 2 \). \( L_{4}= \) Submit Question

To partition the interval \( [-4,8] \) into 4 equal subintervals, we first determine the length of each subinterval. The total length of the interval is \( 8 - (-4) = 12 \), and dividing that by 4 gives us a subinterval length of \( 3 \). The four subintervals are: 1. \( [-4, -1] \) 2. \( [-1, 2] \) 3. \( [2, 5] \) 4. \( [5, 8] \) For the left Riemann sum \( L_4 \), we take the left endpoint of each subinterval: - For the first subinterval, the left endpoint is \( -4 \). - For the second subinterval, the left endpoint is \( -1 \). - For the third subinterval, the left endpoint is \( 2 \). - For the fourth subinterval, the left endpoint is \( 5 \). Thus, the left Riemann sum can be expressed as: \[ L_{4} = \cos(-4) \cdot 3 + \cos(-1) \cdot 3 + \cos(2) \cdot 3 + \cos(5) \cdot 3 \] For the right Riemann sum \( R_4 \), we take the right endpoint of each subinterval: - For the first subinterval, the right endpoint is \( -1 \). - For the second subinterval, the right endpoint is \( 2 \). - For the third subinterval, the right endpoint is \( 5 \). - For the fourth subinterval, the right endpoint is \( 8 \). Thus, the right Riemann sum can be expressed as: \[ R_{4} = \cos(-1) \cdot 3 + \cos(2) \cdot 3 + \cos(5) \cdot 3 + \cos(8) \cdot 3 \]