The area (in sq. units) of the region $\{(x, y): 0 \leq y \leq 2|x|+1, 0 \leq y \leq x^2+1, |x| \leq 3\}$ is

If $f(x)$ is a continuous,increasing,and odd function such that $\int_{-1}^{4} f(x) \,dx = 10$ and $\int_{0}^{1} f(x) \,dx = \frac{3}{2}$,then the area bounded by $y = f(x)$,the $x$-axis,and the ordinates $x = -4$ and $x = 4$ is:

Let $P_{1}: y = 4x^{2}$ and $P_{2}: y = x^{2} + 27$ be two parabolas. If the area of the bounded region enclosed between $P_{1}$ and $P_{2}$ is six times the area of the bounded region enclosed between the line $y = \alpha x, \alpha > 0$ and $P_{1}$,then $\alpha$ is equal to:

The area (in sq units) bounded by the curves $x = -2y^2$ and $x = 1 - 3y^2$ is

If the area of the region $\{(x, y): |4-x^2| \leq y \leq x^2, y \leq 4, x \geq 0\}$ is $\left(\frac{80 \sqrt{2}}{\alpha}-\beta\right)$,where $\alpha, \beta \in \mathbb{N}$,then $\alpha+\beta$ is equal to . . . . . . .

Find the area of the region bounded by the parabola $(y-2)^2 = x-1$,the tangent to the parabola at the point $(2,3)$,and the $x$-axis.