Questions: Draw rectangles associated with Riemann sums that converge to the integral below. Use n=4 and 10 subintervals of equal length. ∫ from 0 to 1 (1-x) dx = 1/2 Choose the correct graph of the left Riemann sum with n=4 subintervals of equal length. A. B. C. Choose the correct graph of the left Riemann sum with n=10 subintervals of equal length. A. B. C.

Solution Steps

The integral is from $x=0$ to $x=1$. With $n=4$ subintervals, the width of each rectangle is $\Delta x = \frac{1-0}{4} = \frac{1}{4} = 0.25$.

For the left Riemann sum, the height of each rectangle is determined by the function value at the left endpoint of the subinterval.

• Rectangle 1: $f(0) = 1-0 = 1$
• Rectangle 2: $f(0.25) = 1-0.25 = 0.75$
• Rectangle 3: $f(0.5) = 1 - 0.5 = 0.5$
• Rectangle 4: $f(0.75) = 1 - 0.75 = 0.25$

The graph in option B correctly illustrates this.

With $n=10$ subintervals, the width of each rectangle is $\Delta x = \frac{1-0}{10} = \frac{1}{10} = 0.1$.

The height of each rectangle is determined by the function value at the left endpoint of each subinterval. Since it is a Left Riemann sum, the heights of the rectangles should be $f(0)$, $f(0.1)$, $f(0.2)$, ..., $f(0.9)$. Since $f(x) = 1-x$, the heights will decrease linearly.

The correct graph is B.

Final Answer: The correct graphs are B and B.