AP Calculus AB Riemann Sums and the Definite Integral Quiz: 10 Questions with Explanations

A the slope of a tangent lineB the signed area between a graph and the x-axisC the limit of a derivativeD the value of f at a point

Adding thin rectangle areas approximates the area under the curve, which the definite integral computes exactly.

A \(14\)B \(9\)C \(3\)D \(5\)

Left endpoints 0, 1, 2 give heights 0, 1, 4 with width 1: total 5. The right sum would be 14.

A an underestimateB exactC equal to the left sumD an overestimate

Each right endpoint gives the largest value on its subinterval, so the rectangles stick out above the curve.

A the average of the left and right sumsB the midpoint sumC the left sumD twice the right sum

Each trapezoid has area (f(left) + f(right))/2 times the width, so the total is the average of the two sums.

A \(54\)B \(9\)C \(27\)D \(13.5\)

The triangle has base 6 and height 9, so area (1/2)(6)(9) = 27. Forgetting the one half gives 54.

A \(-6\)B \(6\)C \(0\)D \(\frac{1}{6}\)

Reversing the limits changes the sign of the integral.

A \(f(3)\)B \(0\)C \(1\)D undefined

An interval of zero width has zero area.

A \(20\text{ m}\)B \(32\text{ m}\)C \(26\text{ m}\)D \(10\text{ m}\)

Left sum: 2(4 + 6) = 20. The right sum would be 2(6 + 10) = 32.

A the derivative of the area functionB the average of f on the intervalC the limit of Riemann sums as n approaches infinityD the slope of the secant line

The integral is the limit of the Riemann sums, which gives signed area.

A \(\frac{1}{2}\)B \(0\)C \(2\)D \(-1\)

The graph is odd, so the area above the axis on (0, 1) cancels the area below on (-1, 0).