3.Answer: $x^2 + y^2 = r^2$
4.Answer: A rightward-opening Parabola.
5.Answer: $(0, 0)$ or the Origin.
6.Answer: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a > b$)
7.Answer: A V-shaped curve (opening upwards from the origin).
8.Answer: At $x = 0$, $x = \pi$, and $x = 2\pi$.
9.Answer: Completing the square gives $(x-2)^2 + y^2 = 4$. Center is $(2, 0)$.
10.Answer: Upward (Symmetric about the positive y-axis).
11.Answer: Area = $\int_a^b y \,dx$ or $\int_a^b f(x) \,dx$.
13.Answer: We take the absolute value (or modulus) of the integral. Area = $|\int_a^b y \,dx|$.
14.Answer: Area = $\int_c^d x \,dy$ or $\int_c^d f(y) \,dy$.
15.Answer: $\int_1^3 x^2 \,dx$.
16.Answer: $\int_0^2 x \,dx = [\frac{x^2}{2}]_0^2 = \frac{4}{2} - 0 = 2$ sq units.
17.Answer: $\int_1^4 3 \,dx = [3x]_1^4 = 12 - 3 = 9$ sq units.
18.Answer: $\int_0^{\pi/2} \cos x \,dx$.
19.Answer: Using horizontal strips: $\int_0^3 x \,dy = \int_0^3 2y \,dy = [y^2]_0^3 = 9 - 0 = 9$ sq units.
20.Answer: No. (The integral evaluates to 0 because the positive and negative areas cancel out).
21.Answer: $\int_0^\pi \sin x \,dx + \left| \int_\pi^{2\pi} \sin x \,dx \right|$.
22.Answer: $\int_0^1 e^x \,dx = [e^x]_0^1 = e^1 - e^0 = e - 1$ sq units.
23.Answer: $\int_1^e \frac{1}{x} \,dx = [\ln x]_1^e = \ln e - \ln 1 = 1 - 0 = 1$ sq unit.
24.Answer: $\int_0^4 \sqrt{x} \,dx$.
25.Answer: $[\frac{2}{3}x^{3/2}]_0^4 = \frac{2}{3}(4)^{3/2} = \frac{2}{3}(8) = \frac{16}{3}$ sq units.
26.Answer: Both axes (x-axis and y-axis).
27.Answer: Multiply by 4.
28.Answer: $y = \sqrt{a^2 - x^2}$ (Taking the positive square root).
30.Answer: $y = \frac{b}{a} \sqrt{a^2 - x^2}$.
32.Answer: Multiply by 2.
33.Answer: The First Quadrant.
34.Answer: No (It is symmetric about the origin / opposite quadrants).
36.Answer: Area = $\int_a^b [f(x) - g(x)] \,dx$ or $\int_a^b (y_{upper} - y_{lower}) \,dx$.
37.Answer: By finding the points of intersection of the two curves (equating $f(x) = g(x)$ and solving for $x$).
38.Answer: $x^2 = x \implies x^2 - x = 0 \implies x(x - 1) = 0 \implies x = 0$ and $x = 1$.
39.Answer: $\int_0^1 (x - x^2) \,dx$.
40.Answer: $y = x$ (Since $x > x^2$ for $0 < x < 1$).
41.Answer: Sub $y=x^2$ into $y^2=x \implies (x^2)^2 = x \implies x^4 - x = 0 \implies x(x^3-1)=0 \implies x=0, x=1$. Points are $(0,0)$ and $(1,1)$.
42.Answer: Upper curve is $y = \sqrt{x}$, lower is $y = x^2$. Integral is $\int_0^1 (\sqrt{x} - x^2) \,dx$.
43.Answer: A Line Segment (connecting $(1, \sqrt{3})$ and $(1, -\sqrt{3})$).
44.Answer: Intersections at $x=0, 1$. Upper curve is $\sqrt{x}$. Integral: $\int_0^1 (\sqrt{x} - x) \,dx$.
45.Answer: $x^2 = 4 \implies x = \pm 2$. The points are $x = -2$ and $x = 2$.
48.Answer: Here $a^2=16 \implies a=4$ and $b^2=9 \implies b=3$. Area = $\pi(4)(3) = 12\pi$ sq units.
49.Answer: $r^2=25$. Area = $\pi(25) = 25\pi$ sq units.
50.Answer: $\frac{8a^2}{3}$.
51.Answer: Here $4a = 4 \implies a = 1$. Area = $\frac{8(1)^2}{3} = \frac{8}{3}$ sq units.
52.Answer: $\frac{16ab}{3}$.
53.Answer: $4a = 4 \implies a=1$. $4b = 4 \implies b=1$. Area = $\frac{16(1)(1)}{3} = \frac{16}{3}$ sq units.
54.Answer: Area = $\pi a b \implies \pi(5)(b) = 20\pi \implies 5b = 20 \implies b = 4$.
55.Answer: Compare $y^2 = 8x$ to $y^2 = 4ax \implies 4a=8 \implies a=2$. Compare $x^2 = 8y$ to $x^2 = 4by \implies 4b=8 \implies b=2$. Therefore $a=2, b=2$.
