Vardaan Learning Institute

Find the x-intercepts of the downward-opening parabola $y = 4 - x^2$.

Find the coordinates of the vertex of the parabola $y = x^2 - 2x + 1$.

Find the center and the radius of the circle given by $x^2 + y^2 - 6x = 0$.

Find the points of intersection of the line $y = x$ and the parabola $y = x^2$.

Find the points of intersection of the two parabolas $y^2 = x$ and $x^2 = y$.

Determine if the vertex of the modulus function $y = |x - 2|$ lies on the x-axis or the y-axis, and state its coordinates.

Find the points where the circle $x^2 + y^2 = 4$ intersects the y-axis.

Find the x-coordinate where the curves $y = \sin x$ and $y = \cos x$ intersect in the interval $[0, \frac{\pi}{2}]$.

What is the y-intercept of the exponential curve $y = e^x$?

Find the x-coordinates of the intersection points of the line $y = 2x$ and the parabola $y = x^2$.

Evaluate the area bounded by the line $y = x$, the x-axis, and the ordinates $x = 1$ and $x = 3$.

Find the area of the region bounded by the curve $y = x^2$ and the x-axis from $x = 0$ to $x = 2$.

Calculate the area under the cubic curve $y = x^3$ bounded by the x-axis and the lines $x = 0$ and $x = 1$.

Find the area of the region bounded by the curve $y = \sin x$, the x-axis, and the lines $x = 0$ and $x = \pi$.

Calculate the area bounded by the parabola $x = y^2$, the y-axis, and the horizontal lines $y = 1$ and $y = 2$.

Evaluate the area bounded by the rectangular hyperbola $y = \frac{1}{x}$, the x-axis, and the ordinates $x = 1$ and $x = e^2$.

Find the area bounded by the curve $y = e^x$, the x-axis, and the lines $x = 0$ and $x = 2$.

Calculate the area enclosed by the curve $y = \cos x$, the x-axis, and the ordinates $x = 0$ and $x = \frac{\pi}{2}$.

Find the area bounded by the curve $y = 3x^2$, the x-axis, and the vertical lines $x = 1$ and $x = 2$.

Evaluate the area bounded by the curve $y = \sqrt{x}$, the x-axis, and the lines $x = 0$ to $x = 9$.

Find the area of the region bounded by the parabola $y = x^2$ and the line $y = 4$.

Calculate the area bounded by the parabola $y^2 = x$ and the vertical line $x = 1$.

Find the area of the region bounded by the modulus function $y = |x|$ and the horizontal line $y = 2$.

Set up the definite integral for the area bounded between the parabola $y = x^2 + 1$ and the line $y = x + 1$.

Evaluate the area from the integral set up in Question 24.

Find the area of the region bounded by the parabola $x^2 = 4y$ and the straight line $y = 2$.

Find the area bounded by the curve $y = x|x|$ and the x-axis between $x = -1$ and $x = 1$.

Calculate the area of the triangle formed by the lines $y = x$, $y = -x$, and $y = 3$.

Find the area of the region bounded by the straight line $y = 2x + 1$, the x-axis, and the ordinates $x = 0$ and $x = 2$.

Find the area of the region in the first quadrant bounded by the curves $y = x^3$ and $y = x$.

Set up and evaluate the integral for the area of the circle $x^2 + y^2 = 9$ located strictly in the first quadrant.

Using the concept of symmetry and the result from Q31, what is the total area of the circle $x^2 + y^2 = 16$?

Find the area of the ellipse $\frac{x^2}{4} + \frac{y^2}{9} = 1$ bounded in the first quadrant.

Set up the definite integral using symmetry to find the total area bounded by the parabola $y^2 = 4x$ and the line $x = 3$.

Evaluate the definite integral set up in Question 34.

What is the total area of the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$? (Use symmetry or standard formula).

Find the area of the region bounded by $|x| + |y| = 1$ that lies only in the first quadrant.

Using symmetry, find the total area bounded by the curve $|x| + |y| = 1$.

Find the area of the region bounded by $y = \cos x$ and the x-axis between $x = -\frac{\pi}{2}$ and $x = \frac{\pi}{2}$.

Write the expression for the area enclosed between the circle $x^2 + y^2 = a^2$ and the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a > b$) strictly in the first quadrant.

Use the shortcut formula ($\frac{16ab}{3}$) to find the area bounded between the parabolas $y^2 = 4x$ and $x^2 = 4y$.

Calculate the area bounded between the parabolas $y^2 = 8x$ and $x^2 = 8y$ using the shortcut formula.

The shortcut formula for the area bounded by a parabola $y^2 = 4ax$ and its latus rectum is $\frac{8a^2}{3}$. Apply this to find the area for $y^2 = 12x$.

Calculate the area bounded by the upward parabola $x^2 = 16y$ and its latus rectum.

Find the total area of the region bounded by the ellipse $4x^2 + 9y^2 = 36$.

Find the area of the region bounded by the parabola $y^2 = 4x$ and the line $y = x$.

The shortcut for the area between $y^2 = 4ax$ and $y = mx$ is $\frac{8a^2}{3m^3}$. Verify your answer to Question 46 using this formula.

Use the shortcut formula to find the area bounded by the parabola $y^2 = x$ and the line $y = 2x$.

Find the area of the region defined by $\{(x, y) : x^2 \le y \le x\}$.

Write the expression for the total area enclosed between the standard ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and the auxiliary circle $x^2 + y^2 = a^2$.