Calculating Area $A_1$
Area $A_1$ is enclosed by $y = x^2 + 2$, $x + y = 8$ (or $y = 8 - x$), and the $y$-axis ($x=0$) in the first quadrant.
- Find intersection points: Set $x^2 + 2 = 8 - x$. This gives $x^2 + x - 6 = 0$, or $(x+3)(x-2) = 0$. In the first quadrant, the intersection is at $x=2$.
- Set up the integral: The upper curve is $y = 8 - x$ and the lower curve is $y = x^2 + 2$ between $x=0$ and $x=2$. $A_1 = \int_{0}^{2} [(8 - x) - (x^2 + 2)] dx$
- Calculate the integral: $A_1 = \int_{0}^{2} (6 - x - x^2) dx = \left[ 6x - \frac{x^2}{2} - \frac{x^3}{3} \right]_{0}^{2}$ $A_1 = \left( 6(2) - \frac{2^2}{2} - \frac{2^3}{3} \right) - (0) = 12 - 2 - \frac{8}{3} = 10 - \frac{8}{3} = \frac{30 - 8}{3} = \frac{22}{3}$
Calculating Area $A_2$
Area $A_2$ is enclosed by $y = x^2 + 2$, $y^2 = x$ (or $y = \sqrt{x}$ in the first quadrant), $x = 2$, and the $y$-axis ($x=0$) in the first quadrant.
- Determine the upper and lower curves: For $x \in [0, 2]$, $x^2+2 \ge \sqrt{x}$. Thus, $y = x^2 + 2$ is the upper curve and $y = \sqrt{x}$ is the lower curve.
- Set up the integral: The area is between $x=0$ and $x=2$. $A_2 = \int_{0}^{2} [(x^2 + 2) - \sqrt{x}] dx$
- Calculate the integral: $A_2 = \int_{0}^{2} (x^2 + 2 - x^{1/2}) dx = \left[ \frac{x^3}{3} + 2x - \frac{x^{3/2}}{3/2} \right]_{0}^{2}$ $A_2 = \left[ \frac{x^3}{3} + 2x - \frac{2}{3}x^{3/2} \right]_{0}^{2}$ $A_2 = \left( \frac{2^3}{3} + 2(2) - \frac{2}{3}(2^{3/2}) \right) - (0)$ $A_2 = \frac{8}{3} + 4 - \frac{2}{3}(2\sqrt{2}) = \frac{8}{3} + \frac{12}{3} - \frac{4\sqrt{2}}{3} = \frac{20 - 4\sqrt{2}}{3}$
Calculating $A_1 - A_2$
Subtract the calculated areas $A_1$ and $A_2$. $A_1 - A_2 = \frac{22}{3} - \frac{20 - 4\sqrt{2}}{3}$
- Perform the subtraction: $A_1 - A_2 = \frac{22 - (20 - 4\sqrt{2})}{3} = \frac{22 - 20 + 4\sqrt{2}}{3} = \frac{2 + 4\sqrt{2}}{3}$
- Factor the result: $A_1 - A_2 = \frac{2(1 + 2\sqrt{2})}{3} = \frac{2}{3}(1 + 2\sqrt{2})$
This result matches option C, which is $\frac{2}{3}(2\sqrt{2} + 1)$.
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 $f(\alpha)$ denote the area of the region in the first quadrant bounded by $x = 0, x = 1, y^2 = x$ and $y = |\alpha x - 5| - |1 - \alpha x| + \alpha x^2$. Then $(f(0) + f(1))$ 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 $f$ 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 $f : \mathbf{R} \rightarrow \mathbf{R}$ 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 _________.
If $\int \frac{2x+5}{\sqrt{7-6x-x^2}} dx$ = $A\sqrt{7-6x-x^2} + Bsin^{-1} \left( \frac{x+3}{4} \right) + C$
(Where C is a constant of integration), then the ordered pair (A,B) is equal to :-
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 $f(\alpha)$ denote the area of the region in the first quadrant bounded by $x = 0, x = 1, y^2 = x$ and $y = |\alpha x - 5| - |1 - \alpha x| + \alpha x^2$. Then $(f(0) + f(1))$ 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 __________.
