Integration and Accumulation of Change — High School Calculus practice questions

If F' = f on [a, b], then ∫ₐᵇ f(x) dx equals:

1. F(b) − F(a)
2. f(b) − f(a)
3. F(a) − F(b)
4. F'(b) − F'(a)

Answer: A — F(b) − F(a)

A) Correct — This is the second part of the Fundamental Theorem. B) Uses f instead of its antiderivative F. C) Subtracts in the wrong order. D) Uses derivatives of F, which are just f.

f'(x) = 6x² and f(0) = 5. Find f(x).

1. 2x³ + 5
2. 2x³
3. 12x + 5
4. 6x³ + 5

Answer: A — 2x³ + 5

A) Correct — Antidifferentiate to 2x³ + C, and f(0) = 5 gives C = 5. B) Ignores the initial condition. C) Differentiates instead of integrating. D) Forgets to divide by the new exponent.

Evaluate ∫₀² 3x² dx.

1. 12
2. 8
3. 24
4. 4

Answer: B — 8

B) Correct — The antiderivative is x³: 2³ − 0³ = 8. A) Evaluates the integrand 3x² at 2 instead of the antiderivative. C) Raises the exponent but forgets to divide by 3: 3x³. D) Uses x² as the antiderivative, forgetting to raise the exponent.

Find ∫ x² dx.

1. 2x + C
2. x³/3 + C
3. x³ + C
4. x³/3

Answer: B — x³/3 + C

B) Correct — Raise the exponent by 1 and divide by the new exponent. A) Differentiates instead of integrating. C) Forgets to divide by the new exponent, 3. D) Leaves off the constant of integration, + C.

∫₀³ f(x) dx = 4 and ∫₀⁷ f(x) dx = 10. Find ∫₃⁷ f(x) dx.

1. 14
2. 6
3. −6
4. 2.5

Answer: B — 6

B) Correct — The integrals add: 4 + ∫₃⁷ f = 10. A) Adds the two integrals. C) Subtracts in the wrong order. D) Divides 10 by 4.

f(0) = 1, f(1) = 3, f(2) = 4. Use the trapezoid rule with width 1 to estimate ∫₀² f(x) dx.

1. 8
2. 5.5
3. 4
4. 7

Answer: B — 5.5

B) Correct — (1/2)[1 + 2(3) + 4] = 5.5. A) Adds the three values. C) Is the left Riemann sum, 1 + 3. D) Is the right Riemann sum, 3 + 4.

Find ∫ (3x² − 4x + 1) dx.

1. 6x − 4 + C
2. x³ − 4x² + x + C
3. x³ − 2x² + x + C
4. x³ − 2x² + C

Answer: C — x³ − 2x² + x + C

C) Correct — Integrate term by term. A) Differentiates instead of integrating. B) Forgets to divide 4x by the new exponent 2. D) Drops the antiderivative of 1, which is x.

Evaluate ∫₀^(2π) sin x dx.

1. 4
2. 2
3. 2π
4. 0

Answer: D — 0

D) Correct — The area above the axis on [0, π] cancels the area below it on [π, 2π]. A) Is the total area, which ignores the signs. B) Is the area of only one arch. C) Is the length of the interval.

Find ∫ (1/x) dx.

1. x⁰/0 + C
2. −1/x² + C
3. eˣ + C
4. ln|x| + C

Answer: D — ln|x| + C

D) Correct — The derivative of ln|x| is 1/x. A) Applies the power rule, which fails for x^(−1). B) Differentiates 1/x instead. C) Uses the inverse function of ln.

∫ₐᵇ f(x) dx = 5 and ∫ₐᵇ g(x) dx = −2. Find ∫ₐᵇ [3f(x) − g(x)] dx.

1. 13
2. 7
3. 3
4. 17

Answer: D — 17

D) Correct — 3(5) − (−2) = 17. A) Subtracts 2 instead of −2. B) Forgets to multiply the f integral by 3. C) Adds 5 + (−2).

What is the relationship between differentiation and integration?

They are inverse operations (FTC). Differentiation finds rates; integration accumulates totals.

What is the trapezoidal rule?

Approximation: ∫_a^b f(x)dx ≈ (Δx/2)[f(x₀) + 2f(x₁) + ... + 2f(xₙ₋₁) + f(xₙ)].

What is ∫sec²x dx?

tan x + C.

Compute ∫(2x)/(x² + 1) dx.

u = x² + 1, du = 2x dx → ∫du/u = ln|u| + C = ln(x² + 1) + C.