Assignment 3: Multiple Integrals and Vector Calculus PDF

B.Tech. I-Semester Assignment 3 Multiple Integrals and Vector calculus Q1. Find the values of a and b such that the surface: 𝑎𝑥 2 − 𝑏𝑦𝑧 = (𝑎 + 2)𝑥 and 4𝑥 2 𝑦 + 𝑧 3 = 4 cut orthogonally at (1,-1,2)....

B.Tech. I-Semester Assignment 3 Multiple Integrals and Vector calculus Q1. Find the values of a and b such that the surface: 𝑎𝑥 2 − 𝑏𝑦𝑧 = (𝑎 + 2)𝑥 and 4𝑥 2 𝑦 + 𝑧 3 = 4 cut orthogonally at (1,-1,2). 5 Ans. 𝑎 = 2 𝑎𝑛𝑑 𝑏 = 1 Q2. Find ∇𝜑, if 𝜑 = log(𝑥 2 + 𝑦 2 + 𝑧 2 ). Ans. 2(𝑥𝑰 + 𝑦𝑱 + 𝑧𝑲)/(𝑥 2 + 𝑦 2 + 𝑧 2 ) Q3. Find the directional derivative of 𝜑 = 𝑥 2 𝑦𝑧 + 4𝑥𝑧 2 at the point (1,-2,1) in the direction of 1 the vector 2𝑰 − 𝑱 − 2𝑲. Ans. 12 3. Q4. Find a unit vector normal to the surface 𝑥𝑦 3 𝑧 2 = 4 at the point (-1, -1, 2). 1 Ans. − (𝑰 + 3𝑱 − 𝑲). √11 Q5. Show that vector (−𝑥 2 + 𝑦𝑧)𝑰 + (4𝑦 − 𝑧 2 𝑥)𝑱 + (2𝑥𝑧 − 4𝑧)𝑲 is solenoidal. Q6. If A and B are irrotational, prove that 𝑨 × 𝑩 is solenoidal. Q7. If 𝑭 = (5𝑥𝑦 − 6𝑥 2 )𝑰 + (2𝑦 − 4𝑥)𝑱, evaluate ∫𝑐 𝑭. 𝑑𝑹 along the curve in the xy-plane, 𝑦 = 𝑥 3 from the point (1,1) to (2,8). Ans. 35 Q8. If 𝑭 = 3𝑥𝑦𝑰 − 𝑦 2 𝑱, evaluate ∫𝑐 𝑭. 𝑑𝑹 along the curve in the xy-plane, 𝑦 = 2𝑥 2 from the point (0,0) to (1,2). Ans. -7/6. Q9. If C is a simple closed curve in the xy-plane not enclosing the origin, show that ∫𝑐 𝑭. 𝑑𝑹 = 𝑦𝑰−𝑥𝑱 0, where 𝑭 =. 𝑥 2 +𝑦 2 Q10. Find the angle between the surfaces 𝑥 2 + 𝑦 2 + 𝑧 2 = 9 and 𝑧 = 𝑥 2 + 𝑦 2 − 3 at the point 8 (2, -1, 2). Ans. 𝜃 = cos−1 3√21. Q11. Evaluate ∬𝑆 𝐹⃗ ⋅ 𝑛̂ 𝑑𝑆 where 𝐹⃗ = (𝑥 + 𝑦 2 )𝑖̂ − 2𝑥𝑗̂ + 2𝑦𝑧𝑘̂ and S is the surface of the plane 2𝑥 + 𝑦 + 2𝑧 = 6 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑜𝑐𝑡𝑎𝑛𝑡. Ans. 81 Q12. Evaluate:  xydxdy over the positive quadrant of the circle A Ans. 1 4 8 a x +y =a 2 2 2 4 a 2 ax 4 a 2 ay Q13. Change the order of integration:   0 2 f ( x, y )dxdy. Ans.   0 2 f ( x, y )dydx x y 4a 4a Q14. Evaluate the double integral by changing the order of Ans. 1 integration.   −y e 0 x y dxdy. 1 1− x 1− x − y 1 5 dxdydz Ans. (log 2 − ) Q15. Evaluate:   0 0 0 ( x + y + z + 1)3. 2 8 Q16. Evaluate:  ( x + y 2 + z 2 )dxdydz where V is the volume of Ans. a 5 2 V the cube bounded by the coordinate planes and the planes x= y=z=a 5 𝑥2 Q17. Evaluate ∫0 ∫0 𝑥(𝑥 2 + 𝑦 2 )𝑑𝑥𝑑𝑦. Ans. 18880.2 Q18. Evaluate ∫ ∫𝐴 𝑥𝑦 𝑑𝑥 𝑑𝑦 where A is the domain bounded by x-axis and coordinate 𝑥 = 𝑎4 2𝑎 and the curve 𝑥 2 = 4𝑎𝑦. Ans. 3 Q19.Change the order of integration in the integral 𝑎 √𝑎2 −𝑦 2 𝐼 = ∫−𝑎 ∫0 𝑓(𝑥, 𝑦)𝑑𝑥𝑑𝑦. 𝑎 √𝑎2 −𝑥 2 Ans. ∫0 ∫−√𝑎2−𝑥 2 𝑓(𝑥, 𝑦)𝑑𝑥𝑑𝑦 16 Q20. Show that the area between the parabolas 𝑦 2 = 4𝑎𝑥 and 𝑥 2 =4ay is 𝑎2. 3 1 𝑧 𝑥+𝑧 Q21. Evaluate ∫−1 ∫0 ∫𝑥−𝑧 (𝑥 + 𝑦 + 𝑧)𝑑𝑥𝑑𝑦𝑑𝑧 Ans. 0 Q22. Calculate the volume of the solid bounded by the planes 𝑎3 𝑥 = 0, 𝑦 = 0, 𝑥 + 𝑦 + 𝑧 = 𝑎 𝑎𝑛𝑑 𝑧 = 0. Ans. 6