BE Instrumentation Engineering छमाही २ (इंजीनियरिंग) - University of Mumbai Important Questions for Applied Mathematics 2

Show that the length of curve ${\displaystyle 9a{y}^2=x{(x-3a)}^2 \text{is} 4\sqrt{3}a}$

Find the value of the integral ${\displaystyle {\int }_0^1\frac{x^2}{1+x^3}}$𝒅𝒙 using Simpson’s (𝟑/𝟖)^𝒕𝒉 rule.

Find the value of the integral ${\displaystyle {\int }_0^1\frac{x^2}{1+x^3}}$𝒅𝒙 using Trapezoidal rule

Find the value of the integral ${\displaystyle {\int }_0^1\frac{x^2}{1+x^3}}$𝒅𝒙 using Simpson’s (1/3)^𝒕𝒉 rule.

Find the perimeter of the curve r=a(1-cos 𝜽)

Change the order of integration and evaluate ${\displaystyle {\int }_0^1\frac{{\int }_x^{\sqrt{2-x^{2}x dx dy}}}{\sqrt{x^2+y^2}}}$

Show that ${\displaystyle {\int }_0^a\sqrt{\frac{x^3}{a^3-x^3}}dx=a\frac{\sqrt{x}\gamma \left(\frac{5}{6}\right)}{\gamma \left(\frac{1}{3}\right)}}$

Compute the value of ${\displaystyle {\int }_0^{\frac{\pi }{2}}\sqrt{\sin x+\cos x}dx}$ usingTrapezoidal rule by dividing into six Subintervals.

Compute the value of ${\displaystyle {\int }_0^{\frac{\pi }{2}}\sqrt{\sin x+\cos x}dx}$ using Simpson’s (1/3)^rd rule by dividing into six Subintervals.

Compute the value of ${\displaystyle {\int }_0^{\frac{\pi }{2}}\sqrt{\sin x+\cos x}dx}$ using Simpson’s (3/8)^th rule by dividing into six Subintervals.

Evaluate I = ${\displaystyle {\int }_0^1{\int }_0^{\sqrt{1+x^2}}\frac{dx.dy}{1+x^2+y^2}}$

Change the order of integration of ${\displaystyle {\int }_0^1{\int }_{-\sqrt{2y-y^2}}^{1+\sqrt{1-y^2}}f(x,y)dxdy}$

Change the order of integration ${\displaystyle {\int }_0^a{\int }_{\sqrt{a^2-x^2}}^{x+3a}f(x,y)dxdy}$

Change the order of Integration and evaluate ${\displaystyle {\int }_0^2{\int }_{\sqrt{2y}}^2\frac{x^2}{{\sqrt{x}}^4-4y^2}dxdy}$

Find the area inside the circle r=a sin𝜽 and outside the cardioide r=a(1+cos𝜽 )

Find the volume of the paraboloid ${\displaystyle x^2+y^2=4z}$ cut off by the plane 𝒛=𝟒

Evaluate ${\displaystyle \int \int \int \sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}-\frac{x^2}{c^2}}}$dx dy dz over the ellipsoid ${\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1.}$

Evaluate ${\displaystyle \int \int xy(x-1)dx dy}$ over the region bounded by 𝒙𝒚 = 𝟒,𝒚= 𝟎,𝒙 =𝟏 and 𝒙 = 𝟒

Evaluate ${\displaystyle \int \int \frac{2x{y}^5}{\sqrt{x^2{y}^2-y^4+1}}dxdy}$, where R is triangle whose vertices are (0,0),(1,1),(0,1).

Find the volume enclosed by the cylinder ${\displaystyle y^2=x}$ and ${\displaystyle y=x^2}$ Cut off by the planes z = 0, x+y+z=2.