Step 1: Calculate Area Between Parabolas $P_1$ and $P_2$
Find the intersection points of $P_1: y = 4x^2$ and $P_2: y = x^2 + 27$. $4x^2 = x^2 + 27$ $3x^2 = 27$ $x^2 = 9 \implies x = \pm 3$
The area, Area$_1$, is the integral of the difference between the upper curve ($P_2$) and the lower curve ($P_1$) from $x=-3$ to $x=3$. $Area$_1 = \int_{-3}^{3} ((x^2 + 27) - 4x^2) dx$ $Area$_1 = \int_{-3}^{3} (27 - 3x^2) dx$ Evaluating the integral: $Area$_1 = [27x - x^3]_{-3}^{3}$ $Area$_1 = (27(3) - 3^3) - (27(-3) - (-3)^3)$ $Area$_1 = (81 - 27) - (-81 + 27) = 54 - (-54) = 108$
Step 2: Calculate Area Between Line $y = \alpha x$ and Parabola $P_1$
Find the intersection points of the line $y = \alpha x$ ($\alpha > 0$) and parabola $P_1: y = 4x^2$. $\alpha x = 4x^2$ $x(4x - \alpha) = 0 \implies x = 0 \text{ or } x = \frac{\alpha}{4}$
The area, Area$_2$, is the integral from $x=0$ to $x=\frac{\alpha}{4}$. The line $y = \alpha x$ is the upper curve. $Area$_2 = \int_{0}^{\alpha/4} (\alpha x - 4x^2) dx$ Evaluating the integral: $Area$_2 = [\frac{\alpha x^2}{2} - \frac{4x^3}{3}]_{0}^{\alpha/4}$ $Area$_2 = \frac{\alpha (\alpha/4)^2}{2} - \frac{4(\alpha/4)^3}{3}$ $Area$_2 = \frac{\alpha^3}{32} - \frac{\alpha^3}{48}$ $Area$_2 = \frac{3\alpha^3 - 2\alpha^3}{96} = \frac{\alpha^3}{96}$
Step 3: Relate Areas and Solve for $\alpha$
Given the condition: Area$_1 = 6 \times Area$_2$. $108 = 6 \times \frac{\alpha^3}{96}$ Simplify the equation: $108 = \frac{\alpha^3}{16}$ Solve for $\alpha^3$: $\alpha^3 = 108 \times 16 = 1728$ Take the cube root: $\alpha = \sqrt[3]{1728} = 12$
Similar Questions
Let $[\cdot]$ be the greatest integer function. If $\alpha = \int_{0}^{64} (x^{1/3} - [x^{1/3}]) dx$, then $\frac{1}{\pi} \int_{0}^{\alpha \pi} \left(\frac{\sin^2 \theta}{\sin^6 \theta + \cos^6 \theta}\right) d\theta$ is equal to ________.
- Let $y = y(x)$ be a differentiable function in the interval $(0, \infty)$ such that $y(1) = 2$, and $\lim_{t \rightarrow x} \left( \frac{t^2 y(x) - x^2 y(t)}{x - t} \right) = 3$ for each $x > 0$. Then $2y(2)$ is equal to
- If $f(x)$ satisfies the relation $f(x) = e^x + \int_0^1 (y + xe^x)f(y)dy$, then $e + f(0)$ is equal to _______.
- $6 \int_0^\pi |(\sin 3x + \sin 2x + \sin x)| dx$ is equal to ____________.
- Let $y = y(x)$ be the solution of the differential equation $\sec x \frac{dy}{dx} - 2y = 2 + 3\sin x$, $x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, $y(0) = -\frac{7}{4}$. Then $y\left(\frac{\pi}{6}\right)$ is equal to :
- If $\int_0^1 4 \cot^{-1}(1 - 2x + 4x^2) dx = a \tan^{-1}(2) - b \log_e(5)$, where $a, b \in \mathbf{N}$, then $(2a + b)$ is equal to __________.
- Let the area of the region bounded by the curve $y = \max\{\sin x, \cos x\}$, lines $x = 0$, $x = \frac{3\pi}{2}$, and the $x$-axis be $A$. Then, $A + A^2$ is equal to ___________.
- The number of elements in the set $S = \left\{ x : x \in [0, 100] \text{ and } \int_{0}^{x} t^2 \sin(x-t)dt = x^2 \right\}$ is _____
- If the solution curve $y = f(x)$ of the differential equation
$(x^2-4)y' - 2xy + 2x(4-x^2)^2 = 0$, $x > 2$,
passes through the point (3, 15), then the local maximum value of $f$ is _____ - Let $y = y(x)$ be the solution of the differential equation $x \sin\left(\frac{y}{x}\right) dy = \left(y \sin\left(\frac{y}{x}\right) - x\right) dx$, $y(1) = \frac{\pi}{2}$ and let $\alpha = \cos\left(\frac{y(e^{12})}{e^{12}}\right)$. Then the number of integral value of $p$, for which the equation $x^2 + y^2 - 2px + 2py + \alpha + 2 = 0$ represents a circle of radius $r \leq 6$, is _________.
Important Questions from Calculus
Let $[\cdot]$ be the greatest integer function. If $\alpha = \int_{0}^{64} (x^{1/3} - [x^{1/3}]) dx$, then $\frac{1}{\pi} \int_{0}^{\alpha \pi} \left(\frac{\sin^2 \theta}{\sin^6 \theta + \cos^6 \theta}\right) d\theta$ is equal to ________.
- Let $y = y(x)$ be a differentiable function in the interval $(0, \infty)$ such that $y(1) = 2$, and $\lim_{t \rightarrow x} \left( \frac{t^2 y(x) - x^2 y(t)}{x - t} \right) = 3$ for each $x > 0$. Then $2y(2)$ is equal to
- If $f(x)$ satisfies the relation $f(x) = e^x + \int_0^1 (y + xe^x)f(y)dy$, then $e + f(0)$ is equal to _______.
- $6 \int_0^\pi |(\sin 3x + \sin 2x + \sin x)| dx$ is equal to ____________.
- Let $y = y(x)$ be the solution of the differential equation $\sec x \frac{dy}{dx} - 2y = 2 + 3\sin x$, $x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, $y(0) = -\frac{7}{4}$. Then $y\left(\frac{\pi}{6}\right)$ is equal to :
