Given the region defined by:
\(x - 2y + 4 \geq 0, \quad x + 2y^2 \geq 0, \quad x + 4y^2 \leq 8, \quad y \geq 0\)
We need to find the area \( A \) of this region and express it in the form \( \frac{m}{n} \) where \( m \) and \( n \) are coprime numbers.
Step 1. Setting Up the Integral: The area is given by:
\(A = \int_0^{\sqrt{8}} \left[ (8 - 4y^2) - (-2y^2) \right] \, dy + \int_{\sqrt{8}}^2 \left[ (8 - 4y^2) - (2y - 4) \right] \, dy\)
Step 2. Evaluating the First Integral:
\(\int_0^{\sqrt{2}} \left[ (8 - 4y^2) - (-2y^2) \right] \, dy = \int_0^{\sqrt{2}} (8 - 2y^2) \, dy\)
Integrating term by term:
\(\int_0^{\sqrt{2}} (8 - 2y^2) \, dy = \left[ 8y - \frac{2y^3}{3} \right]_0^{\sqrt{2}}\)
Substituting the limits:
\(= \left( 8 \times \sqrt{2} - \frac{2 \cdot (\sqrt{2})^3}{3} \right) - (0 - 0) = \frac{16\sqrt{2}}{3}\)
Step 3. Evaluating the Second Integral:
\(\int_{\sqrt{2}}^2 \left[ (8 - 4y^2) - (2y - 4) \right] \, dy\)
Simplifying the integrand:
\(= \int_{\sqrt{2}}^2 (8 - 4y^2 - 2y + 4) \, dy = \int_{\sqrt{2}}^2 (12 - 4y^2 - 2y) \, dy\)
Integrating term by term:
\(= \left[ 12y - \frac{4y^3}{3} - y^2 \right]_{\sqrt{2}}^2\)
Substituting the limits:
\(= \left( 24 - \frac{32}{3} - 4 \right) - \left( 12\sqrt{2} - \frac{16\sqrt{2}}{3} - 2 \right) = \frac{107}{12}\)
Step 4. Final Area Calculation: The total area is:
\(A = \frac{16\sqrt{2}}{3} + \frac{107}{12}\)
Expressing \( A \) in the form \( \frac{m}{n} \) where \( m \) and \( n \) are coprime, we have \( m + n = 119 \).
The shortest distance between the curves $ y^2 = 8x $ and $ x^2 + y^2 + 12y + 35 = 0 $ is:
Let a line passing through the point $ (4,1,0) $ intersect the line $ L_1: \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} $ at the point $ A(\alpha, \beta, \gamma) $ and the line $ L_2: x - 6 = y = -z + 4 $ at the point $ B(a, b, c) $. Then $ \begin{vmatrix} 1 & 0 & 1 \\ \alpha & \beta & \gamma \\ a & b & c \end{vmatrix} \text{ is equal to} $
Resonance in X$_2$Y can be represented as
The enthalpy of formation of X$_2$Y is 80 kJ mol$^{-1}$, and the magnitude of resonance energy of X$_2$Y is: