Application of ...

The area enclosed by $$x^2 + y^2 = 4, y = x^2 + x + 1,$$  $$y=\left [ \sin^{2}\displaystyle \frac{x}{4}+\cos\displaystyle \frac{x}{4} \right ]$$ and $$x$$-axis (where $$[.]$$ denotes the greatest integer function) is:

The area bounded by the function $$f(x)=x^{2}:R^{+}\rightarrow R^{+}$$ and its inverse function is:

Find the area of the region bounded by the curves $$y= log_{e}x $$, $$ y=\sin ^{4} \pi x $$, $$x=0 $$

Find the area bounded by the curves $$\displaystyle\ y=\sqrt{1-x^{2}}$$ and $$\displaystyle\ y=x^{3}-x $$. Also find the ratio in which the y-axis divide this area

Find the area of the region enclosed between the two circles $$\displaystyle\ x^{2}+y^{2}=1$$ & $$(x-1)^{2}+y^{2}=1$$

A polynomial function f(x) satisfies the condition $$f(x+1) =f(x)+2x+1$$. Find $$f(x)$$ if $$f(0)=1$$. Find also the equations of the pair of tangents from the origin on the curve $$y=f(x)$$ and compute the area enclosed by the curve and the pair of tangents.

Find the area enclosed the curves : $$y=ex\log { x } $$ and $$\displaystyle y=\frac { \log { x }  }{ ex } $$ where $$\log { e } =1$$

The area included between the curve $${ x }^{ 2 }+{ y }^{ 2 }={ a }^{ 2 }$$ and $$\displaystyle \sqrt { \left| x \right|  } +\sqrt { \left| y \right|  } =\sqrt { a } \left( a>0 \right) $$ is:

Sketch the region bounded by the curves $$ \displaystyle y=x^{2}$$ & $$ \displaystyle\ y= 2/(1+x^{2})$$. Find the area:

The ratio in which the area bounded by the curves $$y^{2}=4x$$ and $$x^{2}=4y$$ is divided by the line $$x=1$$ is