Application of Integrals - Class 12

Let $f(x) = e^{[x]} \int_{0}^{x} \text{sgn}(\sin(\pi t)) dt$, where $[x]$ denotes the greatest integer less than or equal to $x$. Evaluate the definite integral $\int_{0}^{202} f'(x) dx$, where the integration is performed over the points of differentiability of $f(x)$.

Let $w \in \mathbb{C}$ satisfy $|w + 2| \le 1$. If the equations $z^2 + wz + 1 + i = 0$ and $z^2 + iz + w + 1 = 0$ share at least one common root, let $f(x) = x^2 + \text{Re}(w)x + \text{Im}(w)$ for $x \in \mathbb{R}$. Find the area of the region bounded by the curves $y = |f(x)|$ and $y = 2$.

Let $\vec{a} = \sqrt{3}\hat{i} - \hat{j}$ and $\vec{b}(x) = (\sin x)\hat{i} + (\cos x)\hat{j} + (x^2 e^x)\hat{k}$ be two vectors. Consider the integral $I = \int_0^\pi \sqrt{|\vec{a}|^2 |\vec{b}(x)|^2 - |\vec{a} \times \vec{b}(x)|^2} \, dx$ The value of $I$ is:

Let $y = y(x)$ be the continuously differentiable solution to the differential equation $x \frac{dy}{dx} = y + \sqrt{x^2 - y^2}, \quad x \ge 1$ satisfying the initial condition $y(1) = 0$. Find the value of the integral $\int_{1}^{e^\pi} y(x) \, dx$.

Evaluate the integral $I = \int_{0}^{\sqrt{3}} \frac{1}{(1+x^2)^2} \arcsin\left(\frac{2x}{1+x^2}\right) dx$

Consider the parametric curve defined by $x(t) = t^2 - 2t$ and $y(t) = |t|(3 - t)$ for $t \in \mathbb{R}$. The curve intersects itself at a single point, forming a closed loop. The area of the region enclosed by this loop is:

Let $m$ and $M$ be the global minimum and global maximum values of the function $f(x) = x^3 - 3x^2 + 2$ on the interval $[-1, 4]$. Let $A(t)$ be the area of the region in the $xy$-plane bounded by the curve $y = 2x^3 + 2x - 2$, the $y$-axis, and the horizontal line $y = t$ (for $t \ge -2$). Find the value of $15 \int_{m}^{M} (A'(t))^2 \, dt$

Let $A(1, 2, 3)$, $B(2, -1, 1)$, and $C(-1, 1, 2)$ be three points in $\mathbb{R}^3$. The planar triangle $ABC$ is rotated by an angle of $2\pi$ about the line $L: x = y = z$ to generate a solid of revolution. If the volume of this solid is $V$, then $V$ is equal to:

Consider the improper iterated integrals over the unbounded region $[1, \infty) \times [1, \infty)$: $I_1 = \int_1^\infty \left( \int_1^\infty \frac{x^2-y^2}{(x^2+y^2)^2} \, dy \right) \, dx$ $I_2 = \int_1^\infty \left( \int_1^\infty \frac{x^2-y^2}{(x^2+y^2)^2} \, dx \right) \, dy$ Which of the following represents the ordered pair $(I_1, I_2)$?

Let $C$ be the locus of points $(x,y)$ defined parametrically by $x = 2\sin t$ and $y = 2\cos 2t + 2$ for $t \in \left[0, \frac{3\pi}{2}\right]$. Find the area of the region bounded by $C$ and the x-axis.

Let $R = \left\{ (x, y) \in \mathbb{R}^2 : -10 \le y \le 14, \min(f(y), g(y)) \le x \le \max(f(y), g(y)) \right\}$, where $f(y) = \text{sgn}(y) \sqrt{\{y\} - \{y\}^2}$ and $g(y) = \frac{1}{2} - \left| \{y\} - \frac{1}{2} \right|$. Here, $\{y\}$ denotes the fractional part of $y$ and $\text{sgn}(y)$ denotes the signum function. The area of the region $R$ is:

Let $D$ be the region in the first quadrant bounded by the $y$-axis and the curves $y = \tan^{-1} x$ and $y = \cot^{-1} x$. Evaluate the double integral: $I = \iint_D \frac{x}{\sin^4 y + \cos^4 y} \, dx \, dy$

Let $C_1$ be the parabola $y^2 = 4(x+2)$. Let $C_2$ be a continuous curve passing through the point $(7,6)$ such that for all points $(x,y)$ on $C_2$ with $x \neq 1$, the slope of its normal is given by $\frac{y}{1-x}$. If $A$ is the area of the region bounded by $C_1$ and $C_2$, then the value of $A$ is

Let $C$ be the curve defined by the equation $\ln x = y \ln y + \ln(1 + \ln y)$. Let $R$ be the finite region in the first quadrant bounded by the curve $C$, the $y$-axis, and the line $y=1$. The area of the region $R$ is

Let $x_1$ and $x_2$ ($x_1 < x_2$) be the abscissae of the intersection points of the parabola $P: y = 3x^2 - \frac{3}{2}$ and the hyperbola $H: 4y^2 - 4x^2 = 5$ that lie in the region $y < 0$. Evaluate the integral $I = \int_{0}^{1} \left| x^2 - x_2^2 \right| \, dx - 5x_2^5$

Consider the curves $C_1$ and $C_2$ defined parametrically by $C_1: x = t + \sin t, y = \cos t$ and $C_2: x = t + \sin t, y = \cos 2t$. Let $P$ be the point of intersection of $C_1$ and $C_2$ on the $y$-axis, and $Q$ be their next point of intersection with $x > 0$. The area of the region bounded by $C_1$ and $C_2$ between $P$ and $Q$ is:

The curves $C_1: \log_2(x-2) + \log_2(y-1) = 2$ and $C_2: x - y = 1$ intersect at a real point $P$. A third curve $C_3$ is defined implicitly by the equation $x^3 + y^3 - 5xy - 2x - 6y = 5$ Find the slope of the tangent to the curve $C_3$ at the point $P$.

Let $P_1$ be the plane passing through the point $(1, 1, 1)$ and the line of intersection of the planes $x + y + z = 1$ and $x - y + z = 3$. Let $C$ be the solid cylinder $x^2 + y^2 \le 4$. The plane $P_1$ intersects the cylinder $C$ in an elliptical region $R$. If $A$ is the area of the orthogonal projection of the region $R$ onto the plane $x + y + \sqrt{2}z = 0$, then find $A$.

Evaluate the definite integral: $\int_{0}^{2} \left| |x^2 - 2| - 1 \right| \, dx$

Let $S$ be the region in the $xy$-plane defined by $S = \left\{ (x,y) : x^2 + y^2 \le 4 \text{ and } (y - [x])(x^3 - x^2 - x + 1) \le 0 \right\}$ where $[x]$ denotes the greatest integer less than or equal to $x$. The area of the region $S$ is:

Let $E$ be the ellipse $x^2 + 3y^2 = 12$. The polar of the point $P(6, 2)$ with respect to $E$ is the line $L$. Let $L$ be represented by the equation $f(x,y) = 0$, where $f(x,y)$ is a linear polynomial in $x$ and $y$ with coprime integer coefficients such that $f(0,0) > 0$.

Consider the region $\mathcal{R} = \{ (x,y) \in \mathbb{R}^2 : x^2 + 3y^2 \le 12 \text{ and } f(x,y) \ge 0 \}$.

The area of the region $\mathcal{R}$ is:

Let $\vec{a}, \vec{b}, \vec{c}$ be three non-coplanar vectors such that $[\vec{a}, \vec{b}, \vec{c}] = \sqrt{3}$. A varying vector $\vec{p}(t)$ is defined as $\vec{p}(t) = (\sin t)\vec{a} + (\sin t - \cos t)\vec{b} + (\cos 2t)\vec{c}$ for $t \in [0, 2\pi]$. Let $V(t)$ be the volume of the parallelepiped formed by the vectors $\vec{a}+\vec{b}$, $\vec{b}+\vec{c}$, and $\vec{p}(t)$. If $S$ is the set of all $t \in [0, 2\pi]$ such that the angle between $\vec{p}(t)$ and $(\vec{a}+\vec{b}) \times (\vec{b}+\vec{c})$ is obtuse, find the value of $\int_S V(t) dt$.

Let $f(x) = \int_0^x |t^2 - 1| dt$. Find the area of the region bounded by the curve $y = x \lfloor f''(x) \rfloor$, the x-axis, and the ordinates $x = 0$ and $x = 2$, where $\lfloor \cdot \rfloor$ denotes the greatest integer function.

Let $\vec{a} = 2\hat{i} + \hat{j} - \hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} + \hat{k}$. A planar region $S$ in $\mathbb{R}^3$ consists of all points whose position vector $\vec{r}$ satisfies $\vec{r} \cdot (\vec{a} \times \vec{b}) = 0$, $|\vec{r} \times \vec{a}| \le 6\sqrt{3}$, and $|\vec{r} \times \vec{b}| \le 9\sqrt{3}$. If $R$ is the orthogonal projection of the region $S$ onto the $xy$-plane, find the value of the integral $\iint_R (x - y)^2 \,dx\,dy$.

Let $R$ be the region in the first quadrant bounded by the hyperbola $y^2 - x^2 = 1$, its asymptote $y = x$, and the $y$-axis. If the region $R$ is revolved about the line $y = x$, the volume of the resulting solid is $V$. Find $V$.

Let $F(x) = \int_{0}^{x} \left( \int_{0}^{y} \text{sgn}(y - 2t) e^{t^2} dt \right) dy$ for $x > 0$. Find the value of $F''(2)$.

Let $I = \int_{0}^{1} \int_{x^2}^{x} \arcsin\left(\sin\left(\frac{\pi y}{x}\right)\right) dy \, dx$. If $I = \frac{p\pi}{q}$, where $p$ and $q$ are coprime positive integers, find the value of $p+q$.

Let $C$ be a curve defined parametrically by $x = t^2, y = 2t$ for $t \in \mathbb{R}$. Let $L$ be the polar line of the point $P(-1, 0)$ with respect to $C$. If $R$ is the finite region bounded by $C$ and $L$, evaluate the integral: $I = \iint_R \frac{y^2}{(x+1)^3 \sqrt{1-x}} \,dx\,dy$

Let $D(x) = \begin{vmatrix} \log_2 2 & \log_2 4 & \log_2 8 \\\\ x^3 & x^2 & x \\\\ \sec^2\left(\frac{\pi}{4}\right) & \tan^2\left(\frac{\pi}{4}\right) & 3 \end{vmatrix}$ and $f(x) = D'(x)$.

Find the area of the region bounded by the curves $y = f(x)$ and $y = -3x^2 + 6x + 3$.

Let $P_1: 3x + 4y + 5z = 14$ and $P_2: 4x + 3y - 5z = 14$ be two planes. Let $\Pi$ be the plane bisecting the obtuse angle between $P_1$ and $P_2$. Let $M(h, k, l)$ be the image of the point $N(2, 0, 0)$ in the plane $\Pi$.

Let $R$ be the region in the $xy$-plane bounded by the parabola $x = y^2 - y$ and the line $x = ky$. Find the area of the region $R$.

Let $S$ be the set of all points $(x, y)$ in the Cartesian plane that satisfy the system of equations: $3x^2 + 4y^2 = 16$ $5x^2 - 2y^2 = 18$ Find the number of integer values of $k \in [1, 10]$ such that the inequality $\frac{k|x| - 12}{|y| - 2} \le 3$ holds true for all points $(x, y) \in S$.

Let $f(x) = x^4 - 6x^3 + 12x^2 - 8x + 3$. If $L$ is the tangent line to the curve $y = f(x)$ at its point of inflection with the smaller $x$-coordinate, then the area of the region bounded by the curve $y = f(x)$ and the line $L$ is

Let $C_1$ be the locus of points $(x,y)$ for $x > 0$ such that the matrix equation $\begin{pmatrix} x & 1 \\ x^2 & x \end{pmatrix} \begin{pmatrix} u \\ v \end{pmatrix} = \begin{pmatrix} y \\ x+2 \end{pmatrix}$ has at least one solution for $(u,v)$. Let $C_2$ be the locus of points $(x,y)$ for $x > 0$ such that the matrix $M = \begin{pmatrix} y & 2 \\ 2 & x \end{pmatrix}$ has $\lambda = 0$ as an eigenvalue. If $R$ is the region in the first quadrant bounded by $C_1$, $C_2$, and the line $x = 1$, find the area of $R$.

Let $A$ be the area of the region bounded by the curves $y = \sin^{-1}(\sin x)$ and $y = \cos^{-1}(\sin x)$ for $x \in [0, 2\pi]$. The value of $A$ is:

Consider the region $R = \{ (x, y) \in \mathbb{R}^2 : |y| \le e^{-|x|} \sin(\pi x), \ -1 \le x \le 1 \}$. The area of $R$ is

Evaluate the definite integral: $I = \int_{0}^{2\pi} \frac{x \sqrt{1 - \cos(2x)}}{\sqrt{2} (1 + \cos^2 x)} dx$

A curve $C_1$ passing through the origin has the property that any ray of light moving parallel to the $x$-axis reflects off $C_1$ and passes through the point $S(1,0)$. Let $C_2$ be an ellipse whose foci are the origin and $S$, and whose major axis has length $2$. If $A$ is the area of the region bounded by $C_1$ and $C_2$ containing the point $\left(\frac{1}{2}, 0\right)$, then $A$ equals:

Let $M$ be the global maximum value of the function $g(t) = \int_0^t 3(1 - s^2) ds$ on the interval $[-2, 2]$. Let $A$ be the area of the region bounded by the curve $y = |x^3 - 3x|$ and the line $y = M$. Find the value of $A$.

Let $M(x) = \begin{pmatrix} x^3 - \lambda x & e^{|x|} \sin x & \cos x \\ -e^{|x|} \sin x & x^3 - \lambda x & \sqrt{1+x^2} \\ -\cos x & -\sqrt{1+x^2} & x^3 - \lambda x \end{pmatrix}$, where $\lambda$ is a real parameter. Let $f(x) = \det(M(x))$. If the rows of $M(x)$ are linearly dependent at $x = 2$, then the value of the integral $\int_{-2}^{2} \left( \frac{f(x)}{2 + \cos x} + \frac{3\lambda}{2} x^2 \right) dx$ is equal to:

Let $f: \mathbb{R} \to \mathbb{R}$ be a strictly increasing continuous function defined as $f(x) = x^3 + \frac{7}{2}x + \int_0^x |t - 2| dt$ If $I = \int_{0}^{247} f^{-1}(x) dx$, then the value of $3I$ is equal to:

Let $f: (-\infty, 3] \to (-\infty, 5]$ be defined by $f(x) = -x^2 + 6x - 4$. If $x_0$ is the unique real root of the equation $f(x) = f^{-1}(x)$, evaluate the integral $I = \int_{-4}^{x_0} f^{-1}(x) dx$

Let $S = \{ z \in \mathbb{C} : |z - \text{Re}(z)| \le \text{Re}(z) e^{\text{Re}(z)}, 0 \le \text{Re}(z) \le 2 \}$. Let $S'$ be the image of $S$ under the transformation $w = z e^{i\pi/2}$, and let $S'' = \{ w \in S' : \text{Re}(w) \ge 0 \}$. If $A$ denotes the area of the region $S''$ in the complex plane, and $B = \int_0^{2e^2} f^{-1}(x) dx$ where $f(x) = x e^x$ for $x \ge 0$, find the value of $3A - B$.

Let $I = \int_{0}^{\pi} \frac{x \cos x}{1 + \sin^2 x} \, dx$. Which of the following statements is correct?

Let $R$ be the region in the first quadrant bounded by the curve $y = \left[ \frac{6}{x} \right]$, the x-axis, and the vertical lines $x = 1$ and $x = 4$, where $[t]$ denotes the greatest integer less than or equal to $t$. If $A$ is the area of the region $R$, then the value of $10A$ is:

Let $R$ be the region in the $xy$-plane defined by the inequality $|x^2 - 2x| + |y - x| \le 1$. The area of the region $R$ is

Let $R$ be the region enclosed by the parametric curve $x(t) = t^3 - 3t$ and $y(t) = (t^2 - 1)^2 (t^2 - 3)$, where the parameter $t$ satisfies the inequality $(t^2 - 1)^2 (t^2 - 3) \le 0$. The area of the region $R$ is

Let $f(x, y)$ be a polynomial function satisfying $f(tx, ty) = t^5 f(x, y)$ for all $t, x, y \in \mathbb{R}$. Given $f(1, y) = 1 - 5y^2 + 4y^4$, evaluate the integral $I = \int_{-3}^{3} \left( |f(x, 1)| + f(x, 1) \right) dx$

Let $f(x) = |x|$ and $g(x) = x|x-2| - 1$. Evaluate the integral $\int_{0}^{3} f(g(x)) \, dx$

Let $M$ be a $3 \times 3$ matrix with real entries such that $\det(M) = 2$, and let $A = \text{adj}(M)$. Let $O$ be the origin $(0,0)$. For any real number $x$, let $P(x)$ be the point $(x, \det(xA))$ and $Q$ be the point $(2, \det(2A))$ in the $xy$-plane. If $S(x)$ denotes the area of the triangle $OP(x)Q$, find the value of the integral $\int_{-1}^{2} S(x) \, dx$

Let $\vec{a}(x)$ and $\vec{b}(x)$ be two continuous vector-valued functions in $\mathbb{R}^3$ such that for all $x \in \mathbb{R}$, $|\vec{a}(x)|^2 = 2x^4 + 1$, $|\vec{b}(x)|^2 = x^2 + 2$, and $\vec{a}(x) \cdot \vec{b}(x) = 2x^2 + x$. Evaluate the integral $I = \int_{-1}^{2} |\vec{a}(x) \times \vec{b}(x)| \, dx$