Riemann sums and the accumulation of area

Review Riemann sums and the accumulation of area for AP Calculus AB and try 2 free AP-style practice questions. 13 questions on this topic in AP Calculus AB · Unit 6 · Integration and Accumulation of Change.

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Question 1

Water flows into a tank at rate $v(t)$ liters per minute, $t$ in minutes.

$t$	$0$	$2$	$4$	$6$	$8$
$v(t)$	$3$	$10$	$6$	$1$	$7$

Using a midpoint Riemann sum with $2$ subintervals of equal width, what is the approximation for $\int_0^{8} v(t)\,dt$ ?

Question 2

Suppose engineers at Northgate model the inflow rate into a reservoir with the function $R(t)$ , in liters per minute, where $t$ is minutes since a pump restart. Selected values are shown.

$t$ (min)	0	4	9	12
$R(t)$ (L/min)	10	8	4	3

If they use a left Riemann sum with the three subintervals indicated by the table, what is the approximation of total inflow, in liters, from $t=0$ to $t=12$ ?

11 more questions on this topic

Full practice for AP Calculus AB · Unit 6 · Integration and Accumulation of Change: timed tests, step-by-step explanations for every answer, and AI grading for free-response questions.

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Answer, then see why

The price of gasoline rises 10% and the quantity demanded falls 5%. What happens to total revenue?

Not quite. The answer is B.

Why the other options are wrong

AI grading for free-response

Write your FRQ answer and get it scored point by point against the AP rubric, with the line from your answer that earned or missed each point.

Question 13 / 4 pts

Part a1 / 1 pts

Point 1: Earned

94% sure

Reason: Correctly identifies demand for gasoline as inelastic.

Evidence: "The price of gasoline rises and the quantity falls only a little."

Part b2 / 3 pts

Point 1: Earned

91% sure

Reason: Graph shows relatively inelastic demand.

Evidence: Draws a steep demand curve.

Point 2: Earned

88% sure

Reason: Correct new equilibrium price.

Evidence: Shifts supply left, P1 → P2.

Point 3: Not earned

82% sure

Reason: With inelastic demand, a price increase raises total revenue.

Evidence: "Revenue goes down."

Riemann sums and the accumulation of area: AP Calculus AB Practice Questions | S&D Academy