Integration Mixed Practice (Ch 11 review)
Year 12 Mathematical Methods Unit 3 · §11E–11J · Mr Wong
Learning intentions.
- Identify which integration tool a question is asking for: signed integral, area, between curves, recognition, or average value
- Set out solutions cleanly in the standard tech-free format
Toolkit
FTC: $\int_a^b f\,dx = [G(x)]_a^b$ · Area = split at $x$-intercepts, take $|\cdot|$ on below-axis pieces · Between curves = $\int (\text{top} - \text{bottom})\,dx$ · Recognition: differentiate the given $f$, rearrange · $\bar y = \dfrac{1}{b-a}\int_a^b f\,dx$.
Section A — Definite integrals and areas (tech-free)
1. Evaluate.
a $\displaystyle\int_0^1 (3x^2 + e^x)\,dx$
b $\displaystyle\int_0^{\pi/4} \sec^2 x \,dx$ (hint: antiderivative is $\tan x$)
c $\displaystyle\int_1^4 \dfrac{1}{\sqrt{x}}\,dx$
d $\displaystyle\int_{-1}^{2} (x^2 - 1)\,dx$
2. Find the total area bounded by $y = x^2 - 1$, the $x$-axis, $x = -1$ and $x = 2$.
3. Find the total area bounded by $y = x^3 - 4x$ and the $x$-axis between $x = -2$ and $x = 2$.
Mixed Practice — continued
Sections B (between curves & recognition) · C (average value & application)
Section B — Between curves & recognition
4. Find the area enclosed by $y = 4 - x^2$ and $y = x + 2$.
5. Find the area enclosed by $y = x^2$ and $y = 4x - x^2$.
6. Let $f(x) = \log_e(x^2 + 1)$.
a Find $f'(x)$.
b Hence find $\displaystyle\int_0^1 \dfrac{x}{x^2+1}\,dx$ exactly.
7. Let $f(x) = \sin^2 x$. Find $f'(x)$, hence find $\displaystyle\int_0^{\pi/2} \sin x \cos x \,dx$.
Section C — Average value & application
8. A particle's velocity (m/s) at time $t$ s is $v(t) = 3t^2$ for $0 \le t \le 4$.
a Find the displacement over $[0, 4]$.
b Find the average velocity over $[0, 4]$.
9. Find the average value of $f(x) = 7\sin(\tfrac{x}{2}) - 3$ on $[0, 4\pi]$. (Hint: how does the interval compare to the period?)
10. The velocity of a car (m/s) is $v(t) = 20 - 2t$ for $0 \le t \le 10$. Find (a) the distance travelled, and (b) the average velocity.