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
- 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 _________
- 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 ______.
Let $f(x) = \int \frac{7x^{10} + 9x^8}{(1 + x^2 + 2x^9)^2} \,dx$, $x > 0$, $\lim_{x \rightarrow 0} f(x) = 0$ and $f(1) = \frac{1}{4}$.
If $A = \begin{bmatrix} 0 & 0 & 1 \\ \frac{1}{4} & f'(1) & 1 \\ \alpha^2 & 4 & 1 \end{bmatrix}$ and $B = \text{adj}(\text{adj } A)$ be such that $|B| = 81$, then $\alpha^2$ is equal to
- Let $[\cdot]$ denote the greatest integer function and $f(x) = \lim_{n \to \infty} \frac{1}{n^3} \sum_{k=1}^n \left[ \frac{k^2}{3^x} \right]$. Then $12 \sum_{j=1}^\infty f(j)$ is equal to __________.
- Let $f(x) = \int \frac{dx}{x^{\left(\frac{2}{3}\right)} + 2x^{\left(\frac{1}{2}\right)}}$ be such that $f(0) = -26 + 24\log_{e}(2)$. If $f(1) = a + b\log_{e}(3)$, where $a, b \in \mathbf{Z}$, then $a + b$ is equal to :
Let $f$ be a differentiable function satisfying $f(x) = 1 - 2x + \int_{0}^{x} e^{(x - t)} f(t) dt$, $x \in \mathbf{R}$ and let $g(x) = \int_{0}^{x} (f(t) + 2)^{15} (t - 4)^{6} (t + 12)^{17} dt$, $x \in \mathbf{R}$. If $\text{p}$ and $\text{q}$ are respectively the points of local minima and local maxima of $g$, then the value of $|\text{p} + \text{q}|$ is equal to _________.
- Let $f(x) = \int \frac{(2 - x^2) \cdot e^x}{(\sqrt{1 + x})(1 - x)^{3/2}} dx$. If $f(0) = 0$, then $f\left(\frac{1}{2}\right)$ is equal to :
- Let $f : \mathbf{R} \rightarrow \mathbf{R}$ be a function such that $f(x) + 3f\left(\frac{\pi}{2} - x\right) = \sin x$, $x \in \mathbf{R}$. Let the maximum value of $f$ on $\mathbf{R}$ be $\alpha$. If the area of the region bounded by the curves $g(x) = x^{2}$ and $h(x) = \beta x^{3}, \beta > 0$, is $\alpha^{2}$, then $30\beta^{3}$ is equal to ___________.
Let $f$ be a twice differentiable function such that $f(x) = \int_{0}^{x} \tan(t - x)dt - \int_{0}^{x} f(t)\tan t dt$, $x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.
Then $f''\left(\frac{\pi}{6}\right) + 12f'\left(-\frac{\pi}{6}\right) + f\left(\frac{\pi}{6}\right)$ is equal to __________- Let $y = y(x)$ be the solution of the differential equation $\left(x^2 - x\sqrt{x^2 - 1}\right)dy + \left(y\left(x - \sqrt{x^2 - 1}\right) - x\right)dx = 0, x \geq 1$. If $y(1) = 1$, then the greatest integer less than $y\left(\sqrt{5}\right)$ is _________.
Important Questions from Integral Calculus
- 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 _________
- 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 ______.
Let $f(x) = \int \frac{7x^{10} + 9x^8}{(1 + x^2 + 2x^9)^2} \,dx$, $x > 0$, $\lim_{x \rightarrow 0} f(x) = 0$ and $f(1) = \frac{1}{4}$.
If $A = \begin{bmatrix} 0 & 0 & 1 \\ \frac{1}{4} & f'(1) & 1 \\ \alpha^2 & 4 & 1 \end{bmatrix}$ and $B = \text{adj}(\text{adj } A)$ be such that $|B| = 81$, then $\alpha^2$ is equal to
- Let $[\cdot]$ denote the greatest integer function and $f(x) = \lim_{n \to \infty} \frac{1}{n^3} \sum_{k=1}^n \left[ \frac{k^2}{3^x} \right]$. Then $12 \sum_{j=1}^\infty f(j)$ is equal to __________.
- Let $f(x) = \int \frac{dx}{x^{\left(\frac{2}{3}\right)} + 2x^{\left(\frac{1}{2}\right)}}$ be such that $f(0) = -26 + 24\log_{e}(2)$. If $f(1) = a + b\log_{e}(3)$, where $a, b \in \mathbf{Z}$, then $a + b$ is equal to :
