1.Find the area of the region $\{(x, y) : y^2 \le 4x, 4x^2 + 4y^2 \le 9\}$.
2.Determine the area bounded by the curve $y = \max(\sin x, \cos x)$ and the x-axis, for $x \in [0, \pi]$.
3.Find the area bounded by the loop of the curve $y^2 = x(x-1)^2$.
4.Calculate the area of the region bounded by $y = \sin^{-1}x$, $y = \cos^{-1}x$, and the x-axis.
5.Find the area of the region bounded by the curve $x^{2/3} + y^{2/3} = a^{2/3}$.
6.Determine the area of the region defined by $A = \{(x, y) : x^2 + y^2 \le 1 \le x + y\}$.
7.Find the area of the figure bounded by the curves $y = \log_e x$, $y = \log_e|x|$, $y = |\log_e x|$ and $y = |\log_e|x||$.
8.Calculate the area enclosed by the curve $|x| + |y| = 1$.
9.Find the area of the region bounded by the curves $y = x^2$ and $y = \frac{2}{1+x^2}$.
10.Find the area bounded by $y = e^{-|x|}$ and the lines $x = -1$, $x = 1$, and $y = 0$.
11.Determine the area of the region bounded by the curve $y = x \sin x$ and the x-axis between $x = 0$ and $x = 2\pi$.
12.Find the area of the region bounded by the curves $x = y^2 - 2$ and $x = e^y$ and the lines $y = 1$ and $y = -1$.
13.Find the area bounded by the curve $y = x e^{-x^2}$, the x-axis, and the ordinate $x = a$, and then take the limit as $a \to \infty$.
14.Calculate the area of the region bounded by $y = |x - 1| + |x - 2|$ and $y = 3$.
15.Find the area bounded by the parabola $y = x^2 - 4x + 3$ and the tangents drawn to it at the points where it cuts the x-axis.
16.Determine the area of the region enclosed by the curves $y = \sin x$ and $y = \cos x$ between two consecutive points of intersection.
17.Find the area bounded by the curves $y = \sqrt{x}$ and $y = x^3$.
18.Calculate the area of the region defined by $\{(x, y) : |x-y| \le 2 \text{ and } |x+y| \le 2\}$.
19.Find the area of the loop of the curve $ay^2 = x^2(a - x)$ where $a > 0$.
20.Determine the area of the region bounded by the curve $y = \tan x$, $y = \cot x$, and the x-axis in $[0, \frac{\pi}{2}]$.
21.Find the area enclosed by the curve $(y - \arcsin x)^2 = x - x^2$.
22.Calculate the area of the region bounded by $y = \min(x^2, 1-x^2)$ and the x-axis.
23.Find the area bounded by the curve $y = [x]$ and $y = x$ between $x = 0$ and $x = 3$. ($[x]$ is the greatest integer function).
24.Determine the area bounded by $y = x^2 - 1$ and $y = 1 - x^2$.
25.Find the area bounded by $y = 2 - |x|$ and $y = x^2$.
26.Calculate the area of the region bounded by $y = \log_e(x+e)$ and the line $x = \log_e(1/y)$.
27.Find the area bounded by $x^2 + y^2 \le 2ax$ and $y^2 \ge ax$, where $a > 0$.
28.Determine the area of the region enclosed by $y = e^x$, $y = e^{-x}$ and $x = 1$.
29.Find the area bounded by the curve $x = y^2$ and the line $x = 3 - 2y^2$.
30.Find the area of the region bounded by $y = x^3 - x$ and $y = x - x^2$.
31.Calculate the area bounded by the circles $x^2 + y^2 = 16$ and $x^2 + y^2 - 4x = 0$.
32.Find the area enclosed by $y = \sin^{-1}x$, $y = \cos^{-1}x$, and the y-axis.
33.Determine the area of the region $\{(x, y) : y \ge x^2 \text{ and } y \le |x|\}$.
34.Find the area bounded by the curve $y = \{x\}$ (fractional part) and $y = 1$ from $x=0$ to $x=n$, where $n \in \mathbb{N}$.
35.Find the area of the region bounded by the curve $y = e^x$, the line $y = e$, and the y-axis.
36.Calculate the area of the region bounded by $y^2 = 4ax$ and $x^2 + y^2 = 2ax$.
37.Find the area bounded by the ellipse $x^2 + 4y^2 = 4$ and the straight line $x + 2y = 2$ in the first quadrant.
38.Determine the area enclosed by the curves $y = \frac{1}{x^2+1}$ and $y = \frac{x^2}{2}$.
39.Find the area of the region enclosed by $x^2 + y^2 \le a^2$ and $x+y \ge a$, where $a > 0$.
40.Calculate the area of the smaller region cut off by the line $x = \frac{a}{\sqrt{2}}$ from the circle $x^2 + y^2 = a^2$.
41.Find the area bounded by the curve $y = \ln x$ and the curve $y = (\ln x)^2$.
42.Determine the area bounded by $y = |x - 2|$ and $y = \frac{1}{2}x$.
43.Find the area of the region between the curves $y = \sqrt{x}$ and $y = x^2$ rotated about the line $y = x$. (Wait, just find the plane area between them).
44.Calculate the area enclosed by the curve $x^4 + y^4 = a^4$ in the first quadrant. (Express in terms of Gamma or Beta function if necessary).
45.Find the area of the region bounded by the curves $y = 2^x$, $y = -2^x$, $x=0$, and $x=1$.
46.Determine the area bounded by the curve $y = \cos x$ and the line $y = 1 - \frac{2x}{\pi}$.
47.Find the area enclosed by $y = |x^2 - 1|$ and the x-axis.
48.Find the area bounded by $y = \tan x$ and the tangent to the curve at $x = \frac{\pi}{4}$ and the x-axis.
49.Calculate the area enclosed by the curves $y = \frac{x^2}{4}$ and $y = \frac{8}{x^2+4}$.
50.Determine the area bounded by the curve $y = [\frac{x^2}{a^2}]$ and the curve $y = \{\frac{x^2}{a^2}\}$ for $x \in [0, a]$.
51.Find the area bounded by $y = e^x$, $y = \ln x$, $x=1$, and $x=e$.
52.Calculate the area of the region bounded by the curve $y = \sin^{-1}(\sin x)$ and the x-axis for $x \in [0, 2\pi]$.
53.Determine the area of the loop of the curve $x^3 + y^3 = 3axy$. (Folium of Descartes/Leaf).
54.Find the area of the region defined by $1 \le x^2 + y^2 \le 4$ and $y \ge |x|$.
55.Calculate the area of the region $\{(x, y) : y \ge \sqrt{|x+3|}, 5y \le x + 9 \le 15\}$.
56.Find the area of the region bounded by $y = \frac{1}{2}x^2$, $y = \frac{1}{2}(x-2)^2$, and $y = \frac{1}{2}$.
57.Determine the area bounded by the curve $y = x^2 \ln x$, the x-axis, and the line $x = e$.
58.Find the area enclosed by the curve $y^2 = x^2(1-x^2)$.
59.Calculate the area of the region bounded by $y = |x^2 - 4x + 3|$ and the line $y = 3$.
60.Find the area bounded by $y = \{x\}$, $y = [x]$ and $x = n$ (where $n \in \mathbb{N}$).
