Area Calculation for α = 0
The area function $f(\alpha)$ is defined for the region in the first quadrant bounded by $x=0$, $x=1$, $y=\sqrt{x}$ (from $y^2=x$), and $y = |\alpha x - 5| - |1 - \alpha x| + \alpha x^2$.
For $\alpha = 0$, the upper boundary simplifies: $y = |0 \cdot x - 5| - |1 - 0 \cdot x| + 0 \cdot x^2 = |-5| - |1| = 5 - 1 = 4$. The area $f(0)$ integrates the difference between the upper curve ($y=4$) and the lower curve ($y=\sqrt{x}$) from $x=0$ to $x=1$.
Integral Setup: $f(0) = \int_{0}^{1} (4 - \sqrt{x}) \, dx$
Evaluation: $f(0) = \int_{0}^{1} (4 - x^{1/2}) \, dx = \left[ 4x - \frac{2}{3}x^{3/2} \right]_{0}^{1} = \left( 4(1) - \frac{2}{3}(1)^{3/2} \right) - (0) = 4 - \frac{2}{3} = \frac{10}{3}$
Area Calculation for α = 1
For $\alpha = 1$, the upper boundary function is $y = |x - 5| - |1 - x| + x^2$. In the interval $x \in [0, 1]$: $|x - 5| = -(x - 5) = 5 - x$ and $|1 - x| = 1 - x$.
Substituting these simplifies the function to $y = (5 - x) - (1 - x) + x^2 = 4 + x^2$. The area $f(1)$ integrates the difference between the upper curve ($y=4+x^2$) and the lower curve ($y=\sqrt{x}$) from $x=0$ to $x=1$.
Integral Setup: $f(1) = \int_{0}^{1} ((4 + x^2) - \sqrt{x}) \, dx$
Evaluation: $f(1) = \int_{0}^{1} (4 + x^2 - x^{1/2}) \, dx = \left[ 4x + \frac{x^3}{3} - \frac{2}{3}x^{3/2} \right]_{0}^{1} = \left( 4 + \frac{1}{3} - \frac{2}{3} \right) - (0) = 4 - \frac{1}{3} = \frac{11}{3}$
Final Sum: f(0) + f(1)
Summing the calculated areas: $f(0) + f(1) = \frac{10}{3} + \frac{11}{3}$
$f(0) + f(1) = \frac{21}{3} = 7$
The final result is 7.
Similar Questions
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Important Questions from Integral Calculus
- Let the solution curve of the differential equation $x dy - y dx = \sqrt{x^2 + y^2} dx, x > 0$, $y(1) = 0$, be $y = y(x)$. Then $y(3)$ is equal to
- If $\int (\sin x)^{\frac{-11}{2}} (\cos x)^{\frac{-5}{2}} dx = - \frac{p_1}{q_1} (\cot x)^{\frac{9}{2}} - \frac{p_2}{q_2} (\cot x)^{\frac{5}{2}} - \frac{p_3}{q_3} (\cot x)^{\frac{1}{2}} + \frac{p_4}{q_4} (\cot x)^{\frac{-3}{2}} + C$, where $p_i$ and $q_i$ are positive integers with $\gcd(p_i, q_i) = 1$ for $i = 1, 2, 3, 4$ and C is the constant of integration, then $\frac{15 p_1 p_2 p_3 p_4}{q_1 q_2 q_3 q_4}$ is equal to _________
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If $y = y(x)$ satisfies the differential equation
$16(\sqrt{x+ 9\sqrt{x}})(4 + \sqrt{9 + \sqrt{x}}) \cos y \, dy = (1 + 2 \sin y) dx, x > 0$ and $y(256) = \frac{\pi}{2}, y(49) = \alpha$, then $2 \sin \alpha$ is equal to :- Let a differentiable function $f$ satisfy the equation $\int_{0}^{36} f\left(\frac{tx}{36}\right) dt = 4\alpha f(x)$. If $y = f(x)$ is a standard parabola passing through the points (2, 1) and (– 4, $\beta$), then $\beta^\alpha$ is equal to ______.
