1.What is the algebraic equation of the x-axis?
2.What is the algebraic equation of the y-axis?
3.Write the standard equation of a circle with its center at the origin $(0,0)$ and radius $r$.
4.What geometric curve does the equation $y^2 = 4ax$ represent?
5.What are the coordinates of the vertex of the parabola $x^2 = -4ay$?
6.Write the standard equation of an ellipse centered at the origin with its major axis along the x-axis.
7.What geometric shape does the modulus function $y = |x|$ represent when graphed?
8.At what values of $x$ does the curve $y = \sin x$ cross the x-axis in the interval $[0, 2\pi]$?
9.Find the center of the circle given by the equation $x^2 + y^2 - 4x = 0$.
10.Identify the orientation of the parabola $x^2 = 4y$ (Upward, Downward, Left, or Right).
11.Write the integral formula for the area bounded by the curve $y = f(x)$, the x-axis, and the ordinates $x=a$ and $x=b$.
12.If a curve lies entirely below the x-axis, the definite integral $\int_a^b y \,dx$ will evaluate to a (Positive/Negative) value.
13.How do we find the actual geometrical area if the definite integral evaluates to a negative value?
14.Write the integral formula for the area bounded by the curve $x = f(y)$, the y-axis, and the lines $y=c$ and $y=d$.
15.Set up the definite integral to find the area under the curve $y = x^2$ from $x=1$ to $x=3$.
16.Evaluate the area under the straight line $y = x$ from $x=0$ to $x=2$.
17.Find the area under the constant curve $y = 3$ bounded by $x=1$ and $x=4$.
18.Set up the integral for the area bounded by the curve $y = \cos x$, the x-axis, and the lines $x=0$ to $x=\pi/2$.
19.Evaluate the area of the region bounded by the line $x=2y$, the y-axis, and the lines $y=0$ and $y=3$.
20.Does the raw integral $\int_0^{2\pi} \sin x \,dx$ give the *total enclosed area* by the sine wave and the x-axis from $0$ to $2\pi$? (Yes/No)
21.Write the correct integral expression to find the *true total area* bounded by $y = \sin x$ and the x-axis from $x=0$ to $x=2\pi$.
22.Evaluate the area under the exponential curve $y = e^x$ from $x=0$ to $x=1$.
23.Find the area bounded by the hyperbola $y = \frac{1}{x}$, the x-axis, and the lines $x=1$ and $x=e$.
24.Set up the integral for the area bounded by the curve $y = \sqrt{x}$, the x-axis, and the lines $x=0$ and $x=4$.
25.Evaluate the integral set up in Question 24.
26.The circle $x^2 + y^2 = a^2$ is symmetric about which axis?
27.To find the total area of the circle $x^2 + y^2 = 16$, we can find the area in the first quadrant and multiply it by what number?
28.From the equation $x^2 + y^2 = a^2$, express $y$ in terms of $x$ for the first quadrant.
29.The standard ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is symmetric about both the x-axis and the y-axis. (True / False)
30.Express $y$ in terms of $x$ for the first quadrant part of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.
31.The right-ward opening parabola $y^2 = 4ax$ is symmetric about which axis?
32.To find the total area bounded by the parabola $y^2 = 4ax$ and the vertical line $x = a$, we can find the area above the x-axis and multiply it by what number?
33.Which specific quadrant is defined by the conditions $x \ge 0$ and $y \ge 0$?
34.Is the cubic curve $y = x^3$ symmetric about the y-axis? (Yes / No)
35.Due to symmetry, is the area of the circle $x^2+y^2=4$ in the 2nd quadrant equal to its area in the 4th quadrant? (Yes / No)
36.Write the integral formula for the area between an upper curve $y = f(x)$ and a lower curve $y = g(x)$ from $x=a$ to $x=b$.
37.If the limits of integration ($a$ and $b$) are not explicitly given, how do you find them for the area bounded between two curves?
38.Find the x-coordinates of the points of intersection of the parabola $y = x^2$ and the line $y = x$.
39.Set up the definite integral for the area bounded between the curves $y = x$ and $y = x^2$.
40.In the interval from $x=0$ to $x=1$, which curve acts as the "upper curve": $y = x$ or $y = x^2$?
41.Find the points of intersection (both x and y coordinates) of the parabolas $y^2 = x$ and $x^2 = y$.
42.Set up the integral for the area bounded between $y^2 = x$ and $x^2 = y$ in the first quadrant.
43.What is the geometric shape of the intersection between the region bounded by the circle $x^2+y^2=4$ and the line $x=1$? (Line Segment / Point)
44.Set up the integral for the area bounded by the curve $y = \sqrt{x}$ and the line $y = x$.
45.Find the x-coordinates of the points of intersection of the parabola $y = x^2$ and the horizontal line $y = 4$.
46.What is the standard direct formula for the total area of the circle $x^2 + y^2 = r^2$?
47.What is the standard direct formula for the total area of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$?
48.Find the total area of the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ using the direct shortcut formula.
49.Find the total area of the circle $x^2 + y^2 = 25$ using the direct formula.
50.Write the direct formula for the area bounded by the parabola $y^2 = 4ax$ and its latus rectum.
51.Calculate the area bounded by the parabola $y^2 = 4x$ and its latus rectum using the shortcut formula.
52.What is the direct shortcut formula for the area enclosed between the two parabolas $y^2 = 4ax$ and $x^2 = 4by$?
53.Calculate the area enclosed by the parabolas $y^2 = 4x$ and $x^2 = 4y$ using the short-cut formula.
54.The total area of an ellipse is $20\pi$ sq units. If the semi-major axis $a$ is $5$, find the value of the semi-minor axis $b$.
55.The area of the region bounded by the parabolas $y^2 = 8x$ and $x^2 = 8y$ is given by $\frac{16ab}{3}$. By comparing with standard forms, what are the values of $a$ and $b$?
