Practice Problem Set: Riemann Sums and Definite Integrals

What You Should Know Before Starting

A Riemann sum approximates the area between a function and the x-axis by dividing an interval into smaller pieces and adding the areas of rectangles. As the number of rectangles increases, the approximation can approach the exact accumulated area.

For an interval divided into equal subintervals, the width of each rectangle is:

The general Riemann sum has the form:

The only thing that changes from a left, right, or midpoint sum is the point at which the function is evaluated.

Approximation	Sample Point	Formula
Left Riemann Sum	Left endpoint of each subinterval
Right Riemann Sum	Right endpoint of each subinterval
Midpoint Riemann Sum	Midpoint of each subinterval

From Riemann Sums to Definite Integrals

The definite integral is the exact limit of Riemann sums as the number of subintervals approaches infinity:

This is the central connection between approximation and exact accumulation. A Riemann sum gives you an approximation using a finite number of rectangles. The definite integral gives the exact accumulated value.

For a continuous function on , the definite integral represents the net signed area between the graph and the x-axis.

Important: Area vs. Net Area

A definite integral measures signed accumulation. Areas above the x-axis contribute positively, while areas below the x-axis contribute negatively. If a problem asks for total area, you may need to split the interval at x-intercepts and make the below-axis contributions positive.

Practice Problems

Question 1 — Left Riemann Sum

Use a left Riemann sum with 4 equal subintervals to approximate .

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The interval is , so . The left endpoints are . Therefore, .

So the left Riemann sum is .

Question 2 — Right Riemann Sum

Use a right Riemann sum with 4 equal subintervals to approximate .

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The width is , and the right endpoints are . Thus, .

The right Riemann sum is .

Question 3 — Midpoint Sum

Use a midpoint Riemann sum with 4 equal subintervals to approximate .

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The subinterval width is . The midpoints are . Therefore, .

The midpoint approximation is .

Question 4 — Sigma Notation

Write a Riemann sum in sigma notation for on the interval , using equal subintervals and right endpoints.

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The width is . The right endpoint of subinterval is . Therefore the right Riemann sum is .

Question 5 — Limit Definition

Write the definite integral as a limit of Riemann sums using right endpoints.

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Here , , and . Therefore, .

Question 6 — Evaluate a Definite Integral

Evaluate .

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Find an antiderivative: . By the Fundamental Theorem of Calculus, . Therefore the value of the definite integral is .

Question 7 — Interpret an Integral

The velocity of a particle is meters per second for . What does represent, and what is its value?

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The integral of velocity over time gives net change in position. .

Thus the particle's position changes by meters over the interval.

Question 8 — Net Area vs. Total Area

Let . Find the net signed area between the graph and the x-axis on .

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Because is nonnegative on the interval, the net area is simply .

Hence the net signed area is square units.

Question 9 — Fundamental Theorem Connection

If , find .

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By the Fundamental Theorem of Calculus, . Therefore, .

Question 10 — AP-Style Interpretation

A rate function measures the number of gallons of water entering a tank per minute. Explain the meaning of .

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The integral represents the total amount of water added to the tank from minute to minute .

Its units are gallons because .

Worked Challenge Problem

AP-Style Challenge: Compare Approximation Methods

For , compare a left Riemann sum, a right Riemann sum, and a midpoint Riemann sum using 2 equal subintervals.

Step 1: The interval is , so .

Step 2: The left endpoints are , giving .

Step 3: The right endpoints are , giving .

Step 4: The midpoints are , giving .

Step 5: Since is increasing on the interval, the left sum should underestimate the exact integral and the right sum should overestimate it.

Step 6: The exact value is .

Conclusion:

Common Mistakes

Confusing left and right endpoints

For a left sum, use the left endpoint of every rectangle. For a right sum, use the right endpoint. The number of rectangles does not determine which endpoint you use.

Forgetting the width of each rectangle

A Riemann sum is not just a sum of function values. Every function value must be multiplied by , the width of each subinterval.

Using the wrong value of Δx

Always calculate from the entire interval. If the interval is and there are rectangles, then .

Treating a definite integral as ordinary geometric area

A definite integral measures net signed area. If part of the graph lies below the x-axis, that portion contributes negatively.

Ignoring units in contextual problems

AP Calculus questions frequently test interpretation. If a rate is measured in gallons per minute and you integrate with respect to minutes, your answer should be in gallons.

AP Calculus Checklist

Before submitting a Riemann-sum or definite-integral answer, ask yourself:

Did I identify the correct interval?
Did I calculate correctly?
Am I using left, right, or midpoint sample points as requested?
Did I multiply the function value by ?
If I used sigma notation, did I set the index and endpoints correctly?
If I evaluated an integral, did I apply the bounds in the correct order?
Am I reporting net change or total amount, depending on what the problem asks?
Did I include units when the problem is contextual?

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