AP - Calculus AB/BC:: Integration and Accumulation of Change

The definite integral \(\int_a^b f(x)\,dx\) can be interpreted geometrically as:

- **A)** The average value of \(f'(x)\) on \([a,b]\)
- **B)** The slope of the tangent line to \(f\) at \(x=b\)
- **C)** Always a positive quantity regardless of the function's sign
- **D)** The net signed area between the curve \(y=f(x)\) and the x-axis from \(x=a\) to \(x=b\) (area above the axis counted positively, below counted negatively)

Using \(n\) rectangles of equal width to approximate the area under a curve on \([a,b]\) is known as:

- **A)** Implicit differentiation
- **B)** L'Hopital's Rule
- **C)** A Riemann sum
- **D)** The Mean Value Theorem

In a LEFT Riemann sum, the height of each rectangle is determined by:

- **A)** The function's value at the midpoint of each subinterval
- **B)** The maximum value of the function on the entire interval
- **C)** The function's value at the LEFT endpoint of each subinterval
- **D)** The function's value at the RIGHT endpoint of each subinterval

For an INCREASING function, how does the LEFT Riemann sum compare to the true area under the curve?

- **A)** The left sum always overestimates
- **B)** The left sum UNDERESTIMATES the true area, since each rectangle's height (from the left endpoint) is less than the curve's height elsewhere in that subinterval
- **C)** The relationship depends only on the number of rectangles, not the function's behavior
- **D)** The left sum always equals the true area exactly

For an INCREASING function, how does the RIGHT Riemann sum compare to the true area under the curve?

- **A)** The right sum is always negative
- **B)** The right sum always equals the true area exactly
- **C)** The right sum OVERESTIMATES the true area
- **D)** The right sum underestimates the true area

The formula for the TRAPEZOIDAL RULE approximation of \(\int_a^b f(x)\,dx\) using \(n\) equal subintervals of width \(\Delta x\) is:

- **A)** \(\dfrac{\Delta x}{2}\left[f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n)\right]\)
- **B)** \(\Delta x \left[f(x_0) + f(x_1) + \cdots + f(x_n)\right]\)
- **C)** \(\dfrac{f(a)+f(b)}{2}\)
- **D)** \(\Delta x \cdot f\left(\dfrac{a+b}{2}\right)\)

Summation notation \(\sum_{i=1}^n f(x_i)\Delta x\) represents:

- **A)** A general Riemann sum, adding up \(n\) rectangle areas (height times width) to approximate a definite integral
- **B)** An indefinite integral
- **C)** The derivative of \(f\)
- **D)** The value of \(f\) at a single point

The definite integral is formally defined as:

- **A)** \(\int_a^b f(x)\,dx\) has no formal limit-based definition
- **B)** \(\int_a^b f(x)\,dx = f(b) - f(a)\) by definition
- **C)** \(\int_a^b f(x)\,dx = f'(b) - f'(a)\)
- **D)** \(\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i)\Delta x\), the limit of Riemann sums as the number of rectangles approaches infinity