AP Calculus AB / Integration and Accumulation of Change / Question No. 6

6. Using a left Riemann sum with three subintervals \( [0,1] \), \( [1,2] \), and \( [2,3] \), what is the approximation of \[ \int_0^3 (3 - x)(x + 1)\, dx? \]

7.5

11.5

Answer is: option3

Solution:

Each subinterval is of width:

\( \Delta x = 1 \)

Left endpoint of \( [0,1] \) is \( x = 0 \)
Left endpoint of \( [1,2] \) is \( x = 1 \)
Left endpoint of \( [2,3] \) is \( x = 2 \)

Evaluate the function \( f(x) = (3 - x)(x + 1) \) at those points

\[ f(0) = (3 - 0)(0 + 1) = 3 \cdot 1 = 3 \]

\[ f(1) = (3 - 1)(1 + 1) = 2 \cdot 2 = 4 \]

\[ f(2) = (3 - 2)(2 + 1) = 1 \cdot 3 = 3 \]

Left Riemann Sum = \( \Delta x [f(0) + f(1) + f(2)] = 1[3 + 4 + 3] = 10 \)

10 (Option C)

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