ENGINEERING MATHEMATICS – UNIT 5: MULTIPLE INTEGRALS Time: 3 Hours Max Marks:
100
Part A – (2 Marks each) Answer all questions (10 × 2 = 20 marks)
1. Define a double integral. 2. Write the general form of a triple integral. 3. Define the region of
integration for a double integral. 4. State the geometrical interpretation of a double integral. 5. Write
the transformation formula for a double integral in polar coordinates. 6. Evaluate ∫■¹ ∫■■ y dy dx. 7.
What is meant by changing the order of integration? 8. Define the Jacobian of transformation. 9.
Write the formula for area using double integral. 10. Define the volume of a solid using triple
integral.
Part B – (5 Marks each) Answer any FIVE questions (5 × 5 = 25 marks)
1. Evaluate ∫■² ∫■■ (x + y) dy dx. 2. Evaluate ∫■¹ ∫■^{1−x} (x + y)² dy dx. 3. Find the area bounded
by y = x and y = x² using double integral. 4. Evaluate ∫■^π ∫■^{π/2} sin x cos y dy dx. 5. Change the
order of integration and evaluate ∫■¹ ∫_y¹ x²y dx dy. 6. Evaluate the volume bounded by z = 4 − x² −
y² and the xy-plane. 7. Evaluate ∫■■ ∫■^{√(a² − x²)} √(a² − x² − y²) dy dx.
Part C – (11 Marks each) Answer any THREE questions (3 × 11 = 33 marks)
1. (a) Define double and triple integrals. (b) Evaluate ∫■² ∫■^{x/2} (x² + y²) dy dx. 2. Evaluate ■_R xy
dx dy, where R is bounded by x = 0, y = 0, x + y = 1. 3. Change the order of integration and
evaluate ∫■¹ ∫■^{√(1−y²)} x² dx dy. 4. Evaluate the triple integral ■_V (x + y + z) dV over the region
bounded by x = 0, y = 0, z = 0 and x + y + z = 1. 5. Evaluate the volume of the sphere x² + y² + z² =
a² using triple integrals in spherical coordinates.
Part D – (21 Marks) Answer ONE full question (1 × 21 = 21 marks)
1. (a) Derive the formula for the change of variables in a double integral using Jacobian. (b) Use it
to evaluate ■_R (x + y) dx dy, where R is bounded by x = 0, y = 0, x + y = 1 using the
transformation u = x + y, v = x − y.
OR
2. (a) Define triple integral and explain how to evaluate it in cylindrical coordinates. (b) Using this,
find the volume of the region bounded by the paraboloid z = 4 − x² − y² and the xy-plane.
100
Part A – (2 Marks each) Answer all questions (10 × 2 = 20 marks)
1. Define a double integral. 2. Write the general form of a triple integral. 3. Define the region of
integration for a double integral. 4. State the geometrical interpretation of a double integral. 5. Write
the transformation formula for a double integral in polar coordinates. 6. Evaluate ∫■¹ ∫■■ y dy dx. 7.
What is meant by changing the order of integration? 8. Define the Jacobian of transformation. 9.
Write the formula for area using double integral. 10. Define the volume of a solid using triple
integral.
Part B – (5 Marks each) Answer any FIVE questions (5 × 5 = 25 marks)
1. Evaluate ∫■² ∫■■ (x + y) dy dx. 2. Evaluate ∫■¹ ∫■^{1−x} (x + y)² dy dx. 3. Find the area bounded
by y = x and y = x² using double integral. 4. Evaluate ∫■^π ∫■^{π/2} sin x cos y dy dx. 5. Change the
order of integration and evaluate ∫■¹ ∫_y¹ x²y dx dy. 6. Evaluate the volume bounded by z = 4 − x² −
y² and the xy-plane. 7. Evaluate ∫■■ ∫■^{√(a² − x²)} √(a² − x² − y²) dy dx.
Part C – (11 Marks each) Answer any THREE questions (3 × 11 = 33 marks)
1. (a) Define double and triple integrals. (b) Evaluate ∫■² ∫■^{x/2} (x² + y²) dy dx. 2. Evaluate ■_R xy
dx dy, where R is bounded by x = 0, y = 0, x + y = 1. 3. Change the order of integration and
evaluate ∫■¹ ∫■^{√(1−y²)} x² dx dy. 4. Evaluate the triple integral ■_V (x + y + z) dV over the region
bounded by x = 0, y = 0, z = 0 and x + y + z = 1. 5. Evaluate the volume of the sphere x² + y² + z² =
a² using triple integrals in spherical coordinates.
Part D – (21 Marks) Answer ONE full question (1 × 21 = 21 marks)
1. (a) Derive the formula for the change of variables in a double integral using Jacobian. (b) Use it
to evaluate ■_R (x + y) dx dy, where R is bounded by x = 0, y = 0, x + y = 1 using the
transformation u = x + y, v = x − y.
OR
2. (a) Define triple integral and explain how to evaluate it in cylindrical coordinates. (b) Using this,
find the volume of the region bounded by the paraboloid z = 4 − x² − y² and the xy-plane.
