Double integrals and Triple Integrals

Q. The value of the integral ${\int }_0^2{\int }_0^x{e}^{x+y}dydx$ is

1. $\frac{1}{2}(e-1)$
2. $\frac{1}{2}(e^2-1{)}^2$
3. $\frac{1}{2}(e^2-e)$
4. $\frac{1}{2}{(e-\frac{1}{e})}^2$

Q. By a change of variable $x(u,v)=uv,y(u,v)=v/u$ in double integral, the integrand $f(x,y)$ changes to $f(uv,v/u)\varphi (u,v)$. Then, $\varphi (u,v)$ is

1. $2v/u$
2. $2uv$
3. $v^2$
4. $1$

Q. The value of ${\int }_0^3{\int }_0^x(6-x-y)dxdy$ is

1. $13.5$
2. $27.0$
3. $40.5$
4. $54.0$

Q. $f(x,y)$ is a continuous function defined ouver $(x,y)ϵ[0,1]\times [0,1]$. Given the two constraints, $x>y^2$ and $y>x^2$, the volume under $f(x,y)$ is

1. ${\int }_{y=0}^{y=1}{\int }_{x=y^2}^{x=\sqrt{y}}f(x,y)dxdy$
2. ${\int }_{y=x^2}^{y=}{\int }_{x=y^2}^{x=1}f(x,y)dxdy$
3. ${\int }_{y=0}^{y=1}{\int }_{x=0}^{x=1}f(x,y)dxdy$
4. ${\int }_{y=0}^{y=\sqrt{x}}{\int }_{x=0}^{x=\sqrt{y}}f(x,y)dxdy$

Q. The expression $V={\int }_0^H\pi R^2(1-h/H{)}^{2}dh$ for the volume of a cone is equal to

1. ${\int }_0^R\pi R^2(1-h/H{)}^{2}dr$
2. ${\int }_0^R\pi RH{(1-\frac{r}{R})}^{2}dr$
3. ${\int }_0^{H}2\pi rH(1-r/R)dh$
4. ${\int }_0^{H}2\pi rH{(1+\frac{r}{R})}^{2}dr$

Q.

Two circles pass through $(-1,4)$ and their centres lie on $x^2+y^2+2x+4y=4$. If $r_1$ and $r_2$ are maximum and minimum radii and $\frac{r_1}{r_2}=a+b\sqrt{2}$, then the value of $a+b$ is.

Q. If a triangle PQR has vertex points P(2, 0), Q(0, 2) and R(0, 0), then the value of integral $\iint 5ydxdy$ evaluated over the triangle is

1. $\frac{1}{2}$
2. $\frac{5}{2}$
3. $\frac{20}{3}$
4. $\frac{10}{3}$

Q. A surface $S(x,y)=2x+5y-3$ is integrated once over a path consisting of the points that satissfy $(x+1{)}^2+(y-1{)}^2=\sqrt{2}$. The integral evaluates to

1. $17\sqrt{2}$
2. $\frac{17}{\sqrt{2}}$
3. $\frac{\sqrt{2}}{17}$
4. $0$

Q. The value of ${\int }_{x=0}^1{\int }_{y=0}^{x^2}{\int }_{z=0}^y(y+2z)dzdydx$ is

1. $\frac{1}{53}$
   
2. $\frac{2}{21}$
   
3. $\frac{1}{6}$
   
4. $\frac{5}{3}$
   

Q. The value of the integral ${\int }_0^{\pi }{\int }_y^{\pi }\frac{sinx}{x}dxdy$, is equal to

1. 2

Q. The value of ${\int }_0^{\infty }{\int }_0^{\infty }{e}^{-x^2}.e^{-y^2}dxdy$ is

1. $\frac{\sqrt{\pi }}{2}$
2. $\sqrt{\pi }$
3. $\pi$
4. $\frac{\pi }{2}$

Q. Given the shaded triangular region $P$ shown in the figure. What is $\int \int xydxdy$?

1. $\frac{1}{6}$
2. $\frac{2}{9}$
3. $\frac{7}{16}$
4. $1$