Use a Riemann sum to approximate the area under the graph of f(x) on the given interval. Use the left endpoints. f(x) = 3x - 4; 1 ≤ x ≤ 4, n = 6
A) $\frac { 51 } { 4 }$ = 12.75
B) $\frac { 63 } { 4 }$ = 15.75
C) $\frac { 33 } { 4 }$ = 8.25
D) $\frac { 87 } { 4 }$ = 21.75

Question

Quizplus · Accepted Answer

The width of each rectangle ($\Delta x$) is $\frac{4-1}{6} = \frac{1}{2}$. The left endpoints are $x_0 = 1, x_1 = 1.5, x_2 = 2, x_3 = 2.5, x_4 = 3, x_5 = 3.5$. The Riemann sum is $\sum_{i=0}^{5} f(x_i) \Delta x = \frac{1}{2} [f(1) + f(1.5) + f(2) + f(2.5) + f(3) + f(3.5)]$. Substituting $f(x) = 3x - 4$ gives $\frac{1}{2} [(3(1)-4) + (3(1.5)-4) + (3(2)-4) + (3(2.5)-4) + (3(3)-4) + (3(3.5)-4)] = \frac{1}{2} [(-1) + 0.5 + 2 + 3.5 + 5 + 6.5] = \frac{1}{2} [16.5] = 8.25$. Multiplying by the width, $\Delta x = \frac{1}{2}$, was already accounted for in the calculation, so the final answer is $8.25$, which corresponds to choice C, not A as initially indicated. My correction is that the correct choice is C, based on the calculation provided.

Use a Riemann Sum to Approximate the Area Under the Graph