Area under a curve

The value of $\displaystyle\frac{16}{\pi^3}\int_0^{\pi/2} f(x)g(x)\,dx$ is _____.

Let $n \ge 2$ be a natural number and $f : [0,1] \to \mathbb{R}$ be the function defined by

$$f(x) = \begin{cases} n(1 - 2nx) & \text{if } 0 \le x \le \dfrac{1}{2n}\\[2mm] 2n(2nx - 1) & \text{if } \dfrac{1}{2n} \le x \le \dfrac{3}{4n}\\[2mm] 4n(1 - nx) & \text{if } \dfrac{3}{4n} \le x \le \dfrac1n\\[2mm] \dfrac{n}{n-1}(nx - 1) & \text{if } \dfrac1n \le x \le 1 \end{cases}$$

If $n$ is such that the area of the region bounded by the curves $x = 0$, $x = 1$, $y = 0$ and $y = f(x)$ is 4, then the maximum value of the function $f$ is _____.

Consider the functions $f, g : \mathbb{R} \to \mathbb{R}$ defined by
$$f(x) = x^2 + \frac{5}{12} \qquad\text{and}\qquad g(x) = \begin{cases} 2\left(1 - \dfrac{4|x|}{3}\right), & |x| \le \dfrac34,\\[2mm] 0, & |x| > \dfrac34.\end{cases}$$
If $\alpha$ is the area of the region
$$\left\{(x, y) \in \mathbb{R}\times\mathbb{R} : |x| \le \frac34,\ 0 \le y \le \min\{f(x), g(x)\}\right\},$$
then the value of $9\alpha$ is _____.

If a point $A(x,y)$ lies in the region bounded by the $y$-axis, straight lines $2y+x=6$ and $5x-6y=30$, then the probability that $y<1$ is

The area of the bounded region enclosed by the curve $y=3-\left|x-\dfrac12\right|-|x+1|$ and the $x$-axis is :

The odd natural number a, such that the area of the region bounded by $y=1$, $y=3$, $x=0$, $x=y^a$ is $\dfrac{364}3$, is equal to :

Consider a curve $y=y(x)$ in the first quadrant as shown in the figure. Let the area $A_1$ is twice the area $A_2$. Then the normal to the curve perpendicular to the line $2x-12y=15$ does NOT pass through the point.

Consider two straight lines, each of which is tangent to both the circle $x^2 + y^2 = \frac{1}{2}$ and the parabola $y^2 = 4x$. Let these lines intersect at the point $Q$. Consider the ellipse whose center is at the origin $O(0, 0)$ and whose semi-major axis is $OQ$. If the length of the minor axis of this ellipse is $\sqrt{2}$, then which of the following statement(s) is (are) TRUE?

Let the straight line $x=b$ divide the area enclosed by $y=(1-x)^2$, $y=0$, and $x=0$ into two parts $R_1$ $(0\le x\le b)$ and $R_2$ $(b\le x\le1)$ such that $R_1-R_2=\frac14$. Then $b$ equals

Let $f:[-1,2]\to[0,\infty)$ be a continuous function such that $f(x)=f(1-x)$ for all $x\in[-1,2]$. Let $R_1=\displaystyle\int_{-1}^2xf(x)\,dx$, and $R_2$ be the area of the region bounded by $y=f(x)$, $x=-1$, $x=2$, and the $x$-axis. Then

The area bounded by the curve $y=f(x)$ and the lines $x=0$, $y=0$ and $x=t$, lies in the interval

Area of the region bounded by the curve $y=e^x$ and lines $x=0$ and $y=e$ is

The area of the region bounded by the curve $y=f(x)$, the $x$-axis, and the lines $x=a$ and $x=b$, where $-\infty<a<b<-2$, is