Find the area bounded by the curves $y=x$ and $y=x^2$.

Using definite integrals, find the area of the triangular region whose sides have the equations $y=2x+1, y=3x+1 \& x=4.$

Using integration, find the area of the triangular region whose vertices are $\left(-1,1\right), \left(0,5\right) \& \left(3,2\right).$

If the area of the triangle $ABC$, the coordinates of whose vertices are $A\left(4,1\right), B\left(6,6\right) \& C\left(8,4\right)$ is $k$ sq. units, then the value of $k$ is

Let the area bounded by the lines $x+2y=2,y-x=1 \& 2x+y=7$ be $k$ sq. units, then the value of $k$ is

Using integration, find the area of the region bounded by the lines $y=4x+5, y=5-x \& 4y=x+5.$

Determine the area under the curve $y=\sqrt{r^2-x^2}$ included between the lines $x=0$  and $x=r$. Hence, prove that the area of a circle of radius r is$\pi r^2$.

Given curve is $\frac{x^2}{4}+\frac{y^2}{9}=1$, if the area under the curve and above the $x$-axis is $k\mathrm{\pi }$ square units, find the value of $k$.