Mock Test 1 — Application of Integrals

Find the area of the region bounded by the curve $y^2 = x$, the line $y = x - 2$, and the x-axis in the first quadrant.

The area bounded by the curve $y = \sin x$ and the x-axis between the interval boundaries $x = 0$ and $x = 2\pi$ is:

Find the volume of the solid generated by rotating the region bounded by $y = \sqrt{a^2-x^2}$ from $x = -a$ to $x = a$ around the x-axis using the Disk Method.

The area enclosed by the standard circle profile defined parametrically as $x = R\cos t$ and $y = R\sin t$ ($t \in [0, 2\pi]$) evaluates to:

Find the area of the region enclosed between the curve $y = x^2$ and the line $y = 4$.

Find the volume of the solid generated by rotating the region bounded by $y = x^3$, the x-axis, and the line $x = 1$ around the horizontal x-axis.

Calculate the arc length of the curve $y = \frac{2}{3}(x-1)^{3/2}$ evaluated from $x = 1$ to $x = 4$.

Find the area enclosed between the loop of the curve $y^2 = x^2(1-x)$.

Find the volume of the solid generated by rotating the region bounded by $y = 2x - x^2$ and the x-axis around the horizontal x-axis.

Find the area bounded by the curves $y = \ln x$, $y = 0$, and $x = e$ using horizontal strips.

Find the value of the integer parameter $k$ if the area enclosed between the curves $y^2 = kx$ and $x^2 = ky$ is exactly $3$ square units.

Calculate the total number of intersection points located in the first quadrant for the two curve profiles $y = \sin x$ and $y = \cos x$ within the interval $x \in [0, 2\pi]$.

Find the value of the parameter $m$ if the area enclosed between the parabola $y^2 = 16x$ and the line $y = mx$ is exactly $\frac{2}{3}$ square units.

Assertion (A): The area enclosed by the curve $y = f(x)$ and the x-axis between $x = -1$ and $x = 1$ for the odd function $f(x) = x^3$ evaluates to exactly 0.
Reason (R): Integrating an odd function across a symmetric interval centered at the origin always results in 0 due to signed area cancellation, but calculating physical area requires integrating the absolute value $|f(x)|$.

Assertion (A): The volume of the solid generated by rotating the region bounded by $y = x$ and $y = x^2$ around the horizontal x-axis can be calculated as $V = \pi \int_0^1 (x - x^2)^2 \, dx$.
Reason (R): The Washer Method requires subtracting the square of the inner radius from the square of the outer radius ($R_{\text{outer}}^2 - R_{\text{inner}}^2$) before integrating.