Calculus 2 Final Exam With Complete
Solutions 100% Verified!!
Arc Length Formula CORRECT ANSWERS ₀∫ⁿ [1 + ƒ'(x)²]
Volume of Solid of Revolution (Disk) CORRECT ANSWERS π ₀∫ⁿ ƒ(x)²
Volume of Solid of Revolution (Washer) CORRECT ANSWERS π ₀∫ⁿ ƒ(x)² - g(x)²
Surface Area of Solid of Revolution CORRECT ANSWERS 2π ₀∫ⁿ ƒ(x) √[1 + ƒ'(x)²]
* arc length × f(x) × 2π
Work Required CORRECT ANSWERS ₀∫ⁿ F × D
Work Required (Spring/Chain/Rope) CORRECT ANSWERS ₀∫ⁿ kx
*k = spring constant
Work Required (Pumping Fluids) CORRECT ANSWERS ₀∫ⁿ (density)(volume)(distance)
*ie. ₀ ∫ ² (62.4)(25π dy)(9-y)
Integration by Parts CORRECT ANSWERS ∫ u dv = uv - ∫ v du
*for u, pick derivable (gets smaller) function
*for dv, pick integratable (gets smaller) function
Volume of Solid by Slicing CORRECT ANSWERS ₀∫ⁿ Α(x), if ⊥ x-axis
*Α(y) if ⊥ y-axis
*ie. ∫ (x)² dx, ƒ(x) = x², ⊥ x-axis
or
∫ (√y)² dy, ⊥ y-axis
ƒ'(sin⁻¹ x)
*(sin inverse derivative) CORRECT ANSWERS 1 / √(1-x²)
ƒ'(cos⁻¹ x)
*(cos inverse derivative) CORRECT ANSWERS - 1 / √(1-x²)
ƒ'(tan⁻¹ x)
*(tan inverse derivative) CORRECT ANSWERS + x²
ƒ'(cot⁻¹ x)
*(cot inverse derivative) CORRECT ANSWERS - + x²
cos²x → CORRECT ANSWERS ½ (1 + cos 2x)
Solutions 100% Verified!!
Arc Length Formula CORRECT ANSWERS ₀∫ⁿ [1 + ƒ'(x)²]
Volume of Solid of Revolution (Disk) CORRECT ANSWERS π ₀∫ⁿ ƒ(x)²
Volume of Solid of Revolution (Washer) CORRECT ANSWERS π ₀∫ⁿ ƒ(x)² - g(x)²
Surface Area of Solid of Revolution CORRECT ANSWERS 2π ₀∫ⁿ ƒ(x) √[1 + ƒ'(x)²]
* arc length × f(x) × 2π
Work Required CORRECT ANSWERS ₀∫ⁿ F × D
Work Required (Spring/Chain/Rope) CORRECT ANSWERS ₀∫ⁿ kx
*k = spring constant
Work Required (Pumping Fluids) CORRECT ANSWERS ₀∫ⁿ (density)(volume)(distance)
*ie. ₀ ∫ ² (62.4)(25π dy)(9-y)
Integration by Parts CORRECT ANSWERS ∫ u dv = uv - ∫ v du
*for u, pick derivable (gets smaller) function
*for dv, pick integratable (gets smaller) function
Volume of Solid by Slicing CORRECT ANSWERS ₀∫ⁿ Α(x), if ⊥ x-axis
*Α(y) if ⊥ y-axis
*ie. ∫ (x)² dx, ƒ(x) = x², ⊥ x-axis
or
∫ (√y)² dy, ⊥ y-axis
ƒ'(sin⁻¹ x)
*(sin inverse derivative) CORRECT ANSWERS 1 / √(1-x²)
ƒ'(cos⁻¹ x)
*(cos inverse derivative) CORRECT ANSWERS - 1 / √(1-x²)
ƒ'(tan⁻¹ x)
*(tan inverse derivative) CORRECT ANSWERS + x²
ƒ'(cot⁻¹ x)
*(cot inverse derivative) CORRECT ANSWERS - + x²
cos²x → CORRECT ANSWERS ½ (1 + cos 2x)
