\[ f(x)=3 \cos (x), \quad 0 \leqslant x \in \frac{3 \pi}{4} \] evaluate the left Alemann sum with \( n=0 \), takng the sample polnts to be left endpolnts, (Hound your answer to sha decimal places, \( \square \) What does the Riemann sum represent? The Rlemann sum represents the net area of the rectangles with resped to the Jramita Hlustrate with a diagram.

To evaluate the left Riemann sum for the function \( f(x) = 3 \cos(x) \) over the interval \( [0, \frac{3\pi}{4}] \) with \( n = 0 \), we need to follow these steps: ### Step 1: Determine the width of the interval Since \( n = 0 \), we are not dividing the interval into subintervals. Therefore, the width of the interval \( \Delta x \) is simply the length of the interval: \[ \Delta x = \frac{3\pi}{4} - 0 = \frac{3\pi}{4} \] ### Step 2: Identify the left endpoint For \( n = 0 \), the left endpoint \( x_0 \) is the starting point of the interval: \[ x_0 = 0 \] ### Step 3: Evaluate the function at the left endpoint Now we evaluate the function \( f(x) \) at the left endpoint: \[ f(0) = 3 \cos(0) = 3 \cdot 1 = 3 \] ### Step 4: Calculate the left Riemann sum The left Riemann sum \( L \) is given by the product of the function value at the left endpoint and the width of the interval: \[ L = f(x_0) \cdot \Delta x = 3 \cdot \frac{3\pi}{4} \] Now, we will calculate this value: \[ L = \frac{9\pi}{4} \] ### Step 5: Approximate the value To provide a numerical approximation, we can calculate \( \frac{9\pi}{4} \): \[ \pi \approx 3.14159 \implies \frac{9\pi}{4} \approx \frac{9 \cdot 3.14159}{4} \approx 7.06858 \] Rounding to three decimal places, we get: \[ L \approx 7.069 \] ### Final Answer The left Riemann sum is approximately: \[ \boxed{7.069} \] ### What does the Riemann sum represent? The Riemann sum represents the net area of the rectangles formed under the curve of the function \( f(x) = 3 \cos(x) \) over the interval \( [0, \frac{3\pi}{4}] \). In this case, since we are using the left endpoint, the height of the rectangle is determined by the value of the function at the left endpoint, and the width is the length of the interval. ### Diagram Illustration To illustrate this, you would draw the function \( f(x) = 3 \cos(x) \) on the interval from \( 0 \) to \( \frac{3\pi}{4} \) and show a rectangle that starts at \( x = 0 \) with height \( 3 \) and width \( \frac{3\pi}{4} \). The area of this rectangle visually represents the left Riemann sum.