Solutions — Y12 Methods Integration Mixed Practice (11E

Section A — Definite integrals and areas

Q1. Evaluate.

a $e$ $[x^3 + e^x]_0^1 = (1+e)-(0+1)$

b $1$ $[\tan x]_0^{\pi/4} = 1 - 0$

c $2$ $[2\sqrt x]_1^4 = 4 - 2$

d $0$ $[\tfrac{x^3}{3}-x]_{-1}^{2} = \tfrac{2}{3} - \tfrac{2}{3}$ — signed cancellation

2 $x$-intercept of $y=x^2-1$ at $x=1$ inside $(-1,2)$. $\left|\int_{-1}^{1}(x^2-1)\,dx\right| = \tfrac{4}{3}$; $\int_1^2(x^2-1)\,dx = \tfrac{4}{3}$. Total area $=$ $\dfrac{8}{3}$ sq units

3 $y=x(x-2)(x+2)$ has roots at $-2,0,2$. Positive on $(-2,0)$, negative on $(0,2)$, and odd-symmetric. $\int_0^2 (x^3-4x)\,dx = -4$, so each piece has area $4$. Total $=$ $8$ sq units

Section B — Between curves & recognition

4 $4-x^2 = x+2 \Rightarrow x=-2$ or $1$. Test $x=0$: parabola gives $4 >$ line $2$, so parabola on top. $\int_{-2}^{1}(2-x-x^2)\,dx = \tfrac{7}{6}-(-\tfrac{10}{3}) =$ $\dfrac{9}{2}$ sq units

5 $x^2 = 4x-x^2 \Rightarrow 2x^2 = 4x \Rightarrow x=0,2$. Test $x=1$: top $= 3$, bottom $= 1$, so $4x-x^2$ on top. $\int_0^2(4x-2x^2)\,dx = [2x^2-\tfrac{2x^3}{3}]_0^2 = 8 - \tfrac{16}{3} =$ $\dfrac{8}{3}$ sq units

6a $f'(x) =$ $\dfrac{2x}{x^2+1}$

6b $\int_0^1 \tfrac{x}{x^2+1}\,dx = \tfrac{1}{2}[\ln(x^2+1)]_0^1 = \tfrac{1}{2}(\ln 2 - 0) =$ $\dfrac{1}{2}\ln 2$

7 $f'(x) = 2\sin x \cos x$. So $\int \sin x\cos x\,dx = \tfrac{1}{2}\sin^2 x + c$. Eval: $\tfrac{1}{2}[\sin^2 x]_0^{\pi/2} =$ $\dfrac{1}{2}$

Section C — Average value & application

8a Displacement $= \int_0^4 3t^2\,dt = [t^3]_0^4 =$ $64$ m

8b $\bar v = \tfrac{1}{4}\cdot 64 =$ $16$ m/s

9 Period of $\sin(\tfrac{x}{2})$ is $4\pi$ — exactly the interval. Sine averages to $0$, so $\bar y = -3$

10a $v \ge 0$ on $[0,10]$ (since $v(10)=0$). Distance $=\int_0^{10}(20-2t)\,dt = [20t-t^2]_0^{10} =$ $100$ m

10b $\bar v = \tfrac{100}{10} =$ $10$ m/s (also visible as the midpoint of the linear velocity from $20$ to $0$).

Solutions — Integration Mixed Practice (Ch 11 review)

Section A — Definite integrals and areas

Section B — Between curves & recognition

Section C — Average value & application