Triple Integral of r | Multiple Integral with Trig Limits | Final Answer in Terms of a | Part 12

🔵 Triple Integral of r | Limits Involving sin(θ) and a² − r² | Direct Integration Method 📘 Engineering Mathematics | VTU | Multiple Integrals | Module 1 In this video, we evaluate a triple integral in cylindrical-like coordinates using the direct integration method, without switching to spherical or cylindrical systems: ∭rdzdrdθ with variable limits: z∈[0,(a²−r²)/a] r∈[0,2sinθ] θ∈[0,π/] 🧠 What You’ll Learn: How to handle limits involving trigonometric functions and quadratics Applying integration by substitution and standard integral identities 👨‍🎓 Useful For: VTU B.Tech/B.Sc. students under CBCS or NEP schemes Engineering Maths learners handling advanced integration problems Multiple integration topics involving variable bounds 🧾 VTU Syllabus Mapping: BMATS201 – Module 1 BMATM201 – Module 1 BMATC201 – Module 1 BMATE101 – Module 4 📌 Don’t forget to like, share, and subscribe for more Multiple Integrals and VTU exam-focused tutorials! [ 📐 Watch Next ] 1️⃣ Double Integrals (xy, x³y, x²+y²)   🔗    • Double Integrals Solved Step-by-Step | xy,...   2️⃣ Triple Integration | Standard Problem   🔗    • Triple Integration Made Easy | Standard Pr...   3️⃣ Triple ∫(x+y+z) | Variable Limits   🔗    • Triple Integral of x + y + z Fully Solved ...   4️⃣ Triple ∫xyz | Bounded Region   🔗    • Triple Integration Made Easy | xyz Over Bo...   5️⃣ Triple ∫xyz | VTU Expected Q   🔗    • Triple Integral of xyz Fully Solved | Most...   6️⃣ Triple ∫1/√(1−x²−y²−z²)   🔗    • Triple Integral of 1/√(1 − x² − y² − z²) |...   7️⃣ Triple ∫1/√(a²−x²−y²−z²)   🔗    • Triple Integral of 1/√(a² − x² − y² − z²) ...   8️⃣ Triple ∫e^(x+y+z) | Direct Method   🔗    • Triple Integral of e^(x + y + z) | Fully S...   9️⃣ Triple ∫e^(x+y+z) | Log Limits   🔗    • Triple Integral of e^(x + y + z) with Loga...   🔟 Triple ∫1/(x+y+z)³   🔗    • Triple Integral of 1 / (x + y + z)³ | Mult...   1️⃣1️⃣ Triple ∫ Over Curved Region | 8π   🔗    • Triple Integral | Multiple Integral Over C...   1️⃣2️⃣ Triple ∫r | Trig Limits | Final Answer in ‘a’   🔗    • Triple Integral of r | Multiple Integral w...   1️⃣3️⃣ Double ∫e^{−(x²+y²)} | Polar   🔗    • Double Integral of e^{-(x² + y²)} | Solved...   1️⃣4️⃣ Area of Ellipse, Limacon, Tetrahedron   🔗    • Double & Triple Integrals | Area of Ellips...   1️⃣5️⃣ Beta-Gamma Function Relationship   🔗    • Beta and Gamma Functions Relationship Expl...   1️⃣6️⃣ Proof That Γ(1/2) = √π   🔗    • Proof That Gamma(1/2) = √π | Step-by-Step ...   1️⃣7️⃣ Proof: Γ(n+1) = nΓ(n) = n!   🔗    • Proof That Γ(n+1) = nΓ(n) = n! | Gamma Fun...   1️⃣8️⃣ ∫√sinθ × ∫1/√sinθ = π   🔗    • Evaluate ∫√sinθ dθ × ∫1/√sinθ dθ from 0 to...   1️⃣9️⃣ √sinθ, √tanθ, √cotθ Integrals   🔗    • Evaluate ∫√sinθ × ∫1/√sinθ | Plus √tanθ an...   [🕕 Playlists] BMATS201: M1 –    • BMATS201 Module 1 ( Integral Calculus - Mu...   M2 –    • Engineering Maths: Vector Calculus (Gradie...   M3 –    • Vector Space & Linear Transformation          • Vector Space and Linear Transformations   M4 –    • VTU Maths Module-4 (Numerical Methods) | N...   M5 –    • Numerical Methods 2 ( Numerical Solution o...   BMATE201: M1 –    • Engineering Maths: Vector Calculus (Gradie...   M2 –    • Vector Space & Linear Transformation          • Vector Space and Linear Transformations   M3 –    • Bmate201 module 4 ( Numerical Methods 1 )   M4 –    • VTU Maths Module-4 (Numerical Methods) | N...   M5 –    • Numerical Methods 2 ( Numerical Solution o...   BMATC201: M1 –    • BMATC201 Module 1 ( Integral Calculus - Mu...   M2 –    • Engineering Maths: Vector Calculus (Gradie...   M3 –    • Partial Differential Equations   M4 –    • VTU Maths Module-4 (Numerical Methods) | N...   M5 –    • Numerical Methods 2 ( Numerical Solution o...   BMATM201: M1 –    • BMATM201 Module 1 ( Integral Calculus - Mu...   M2 –    • Engineering Maths: Vector Calculus (Gradie...   M3 –    • Partial Differential Equations   M4 –    • VTU Maths Module-4 (Numerical Methods) | N...   M5 –    • Numerical Methods 2 ( Numerical Solution o...   #TripleIntegral #MultipleIntegration #DirectIntegration #EngineeringMathematics #VTUMaths #BMATS201 #BMATM201 #BMATC201 #BMATE101 #IntegralCalculus #MultivariableCalculus #BTechMaths #MathsForEngineers #ExamMaths #MathsTutorial #NEPScheme #CBCSSyllabus #3DIntegration