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Applications of Integration — High School Calculus practice questions

16 multiple-choice questions and 15 flashcards on Applications of Integration, about 11% of the High School Calculus bank. Every one carries a written rationale.

Written and maintained by the CoStudy team · last updated 2026-10-02 · how we write and review questions

What this chapter covers

Applications of Integration is one of 8 chapters in CoStudy's High School Calculus bank, and it holds 16 of the bank's 150 multiple-choice questions — roughly 11% of the total. That proportion is not arbitrary: chapters follow the certifying body's published exam outline, and the number of questions in each is set by that domain's published weight, so the share of your practice time this chapter takes matches the share of the real exam it accounts for.

Studying by chapter is worth doing once you have a diagnostic score. A single overall percentage tells you whether you are close; it does not tell you which domain is dragging. Working a weak chapter in isolation, and re-testing it in isolation, is the fastest way to move a score that has stalled — and it is why the mock exams in CoStudy report by domain rather than as one number.

Free Applications of Integration practice questions

2 questions drawn from this chapter, with the full rationale shown — the controlling principle behind the right answer, and why each wrong option tempts and fails.

Find the area between y = x and y = x² on [0, 1].

  1. 1/2
  2. 1/3
  3. 5/6
  4. 1/6

Answer: D — 1/6

D) Correct — ∫₀¹ (x − x²) dx = 1/2 − 1/3 = 1/6. A) Integrates only the top curve, x. B) Integrates only the bottom curve, x². C) Adds the integrals instead of subtracting.

A particle has v(t) = t − 1 on [0, 3]. What total distance does it travel?

  1. 2.5
  2. 1.5
  3. 2
  4. 3

Answer: A — 2.5

A) Correct — Split at t = 1: |∫₀¹ (t − 1) dt| + ∫₁³ (t − 1) dt = 0.5 + 2. B) Is the displacement; the backward motion on [0, 1] cancels part of it. C) Integrates only over [1, 3]. D) Is the length of the time interval.

Applications of Integration flashcards

4 cards from the 15 in this chapter.

Volume of a solid with known cross sections?

V = ∫_a^b A(x) dx, where A(x) is the area of a cross section. Squares of side s: A = s². Semicircles of diameter s: A = (π/8)s².

What is the Mean Value Theorem for integrals?

If f is continuous on [a, b], there exists c in [a, b] such that f(c) = (1/(b − a)) ∫_a^b f(x) dx.

What is the average value of f on [a, b]?

(1/(b − a)) ∫_a^b f(x) dx.

WORKED EXAMPLE: Area between y = x² and y = 2x on [0, 2].

Step 1: Find intersection points. Set x² = 2x → x² - 2x = 0 → x(x - 2) = 0 → x = 0, 2. These match the given interval. Step 2: Determine which curve is on top. At x = 1: 2x = 2 and x² = 1, so 2x > x² on (0, 2). Step 3: Set up integral: Area = ∫₀² (top - bottom) dx = ∫₀² (2x - x²) dx. Step 4: Find antiderivative: F(x) = x² - x³/3. Step 5: Evaluate at bounds: F(2) = 4 - 8/3 = 12/3 - 8/3 = 4/3. F(0) = 0. Step 6: Subtract: Area = 4/3 - 0 = 4/3. ANSWER: Area = 4/3.

Practise the full chapter

These are a sample. The full Applications of Integration chapter runs 31 items with per-chapter progress tracking, on the web and in the iOS app.

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