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Home Practice Questions AP Calculus AB Integration and Accumulation of Change Question No. 34

34. The graph of a differentiable function \( f \) on the closed interval \([-2, 8]\) is shown in the figure above. Find a trapezoidal approximation of \[ \int_{-2}^{8} f(x) \, dx \] using five subdivisions of length \( \Delta x = 2 \).






Answer is: option3

21

Solution:

  1. \( f(-2) = -1 \)
  2. \( f(0) = 0 \)
  3. \( f(2) = 3 \)
  4. \( f(4) = 4 \)
  5. \( f(6) = 3 \)
  6. \( f(8) = 2 \)

We use the Trapezoidal Rule formula again:

\[ \int_{-2}^{8} f(x) \, dx \approx \frac{\Delta x}{2} \left[f(-2) + 2f(0) + 2f(2) + 2f(4) + 2f(6) + f(8)\right] \]

Since \( \Delta x = 2 \), we get:

\[ = \frac{2}{2} \left[-1 + 2(0) + 2(3) + 2(4) + 2(3) + 2\right] \]

\[ = 1 \cdot \left[-1 + 0 + 6 + 8 + 6 + 2\right] = -1 + 0 + 6 + 8 + 6 + 2 = 21 \]

Option C

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