Module 6 Quiz: Integration
Quiz Instructions
Answer each question, then reveal the answer to check. Aim for at least 7 out of 10 correct.
1. Find the antiderivative: ∫ (4x³ − 2x + 7) dx.
x4 − x² + 7x + C.
2. Evaluate ∫02 (3x² + 1) dx.
[x³ + x]02 = (8 + 2) − 0 = 10.
3. What does a right Riemann sum with n = 3 give for ∫03 x² dx?
Δx = 1. Right endpoints: 1, 2, 3. R3 = 1(1 + 4 + 9) = 14. (Exact: 9.)
4. Evaluate ∫−11 (ex − 1) dx.
[ex − x]−11 = (e − 1) − (e−1 + 1) = e − 1/e − 2 ≈ 0.350.
5. Find d/dx [∫0x t/(t² + 1) dt].
By FTC Part 2: x/(x² + 1).
6. Find d/dx [∫1sin x t² dt].
FTC Part 2 + chain rule: (sin x)² · cos x = sin²x cos x.
7. If ∫05 f(x) dx = 8 and ∫05 g(x) dx = 3, find ∫05 [2f(x) − g(x)] dx.
2(8) − 3 = 13.
8. Evaluate ∫0π/4 sec² x dx.
[tan x]0π/4 = tan(π/4) − tan(0) = 1 − 0 = 1.
9. A particle moves with v(t) = 4 − 2t on [0, 5]. What is the displacement?
∫05 (4 − 2t) dt = [4t − t²]05 = (20 − 25) = −5.
10. True or false: ∫ab f(x) dx is always positive if f is continuous on [a, b].
False. The definite integral computes signed area. If f(x) < 0 on all or part of [a, b], the integral can be negative or zero.