CALCULUS 2 FINAL EXAM REVIEW
QUESTIONS WITH COMPLETE
ANSWERS
Absolutely Convergent - ANSWER-A series ∑ a_n is called Absolutely convergent if the
series of absolute values ∑ |a_n| is convergent
also if the series is absolutely convergent it is also convergent.
Root Test - ANSWER-L=1 (nothing) L<1 (converges) L>1 (diverges)
Ratio Test - ANSWER-L=1 (nothing) L<1 (converges) L>1 (diverges)
Integral Test Remainder Estimate - ANSWER-For integral remainder 1/sqru(n^4+1) <
1/Sq(n^4) = 1/n^2
take integral of last one and b=inf and a=10
The Comparison Test - ANSWER-(i) If ∑bn is convergent and an≤bn for all n, then ∑ an
is also convergent.
(ii) If ∑bn is divergent and an≥bn for all n, then ∑ an is also divergent
.
practice prob - ANSWER-
Washer Method - ANSWER-pi∫(outer radius)²-(inner radius)²
(x-axis)
Shell Method - ANSWER-2pi∫rh
(y-axis)
Integration by Parts - ANSWER-uv-∫vdu
Arc Length Formula - ANSWER-∫√(1+(dy/dx)^2)
∫lnu - ANSWER-ulnu-u
sin2x - ANSWER-2sinxcosx
sin²x - ANSWER-(1-cos2x)/2
cos²x - ANSWER-(1+cos2x)/2
, ∫secxdx - ANSWER-ln|secx+tanx|
Derivative of secx - ANSWER-secxtanx
Derivative of tanx - ANSWER-sec²x
Derivative of cscx - ANSWER--cscxcotx
Derivative of cotx - ANSWER--csc²x
SA Rotating about y-axis - ANSWER-x
Length of Polar Curve - ANSWER-∫√(r²+(dr/d∅)²)d∅
derivative of sinx - ANSWER-cosx
derivative of cosx - ANSWER--sinx
integral of sinx - ANSWER--cosx
integral of cosx - ANSWER-sinx
Sin(A) Sin(B) - ANSWER-1/2[Cos(A-B)-Cos(A+B)]
Cos(A) Cos(B) - ANSWER-1/2[Cos(A-B)+Cos(A+B)]
Sin(A) Cos(B) - ANSWER-1/2[Sin(A-B)+Sin(A+B)]
f average - ANSWER-(1/B-A)∫f(x)dx
Area - ANSWER-b
∫[f(x)-g(x)]dx
a
√(a²-x²) - ANSWER-x=aSinθ
√(x²-a²) - ANSWER-x=aSecθ
√(a²+x²) - ANSWER-x=aTanθ
Surface area - ANSWER-b
S=∫ 2π f(x) √(1+[f'(x)]²) dx
a
QUESTIONS WITH COMPLETE
ANSWERS
Absolutely Convergent - ANSWER-A series ∑ a_n is called Absolutely convergent if the
series of absolute values ∑ |a_n| is convergent
also if the series is absolutely convergent it is also convergent.
Root Test - ANSWER-L=1 (nothing) L<1 (converges) L>1 (diverges)
Ratio Test - ANSWER-L=1 (nothing) L<1 (converges) L>1 (diverges)
Integral Test Remainder Estimate - ANSWER-For integral remainder 1/sqru(n^4+1) <
1/Sq(n^4) = 1/n^2
take integral of last one and b=inf and a=10
The Comparison Test - ANSWER-(i) If ∑bn is convergent and an≤bn for all n, then ∑ an
is also convergent.
(ii) If ∑bn is divergent and an≥bn for all n, then ∑ an is also divergent
.
practice prob - ANSWER-
Washer Method - ANSWER-pi∫(outer radius)²-(inner radius)²
(x-axis)
Shell Method - ANSWER-2pi∫rh
(y-axis)
Integration by Parts - ANSWER-uv-∫vdu
Arc Length Formula - ANSWER-∫√(1+(dy/dx)^2)
∫lnu - ANSWER-ulnu-u
sin2x - ANSWER-2sinxcosx
sin²x - ANSWER-(1-cos2x)/2
cos²x - ANSWER-(1+cos2x)/2
, ∫secxdx - ANSWER-ln|secx+tanx|
Derivative of secx - ANSWER-secxtanx
Derivative of tanx - ANSWER-sec²x
Derivative of cscx - ANSWER--cscxcotx
Derivative of cotx - ANSWER--csc²x
SA Rotating about y-axis - ANSWER-x
Length of Polar Curve - ANSWER-∫√(r²+(dr/d∅)²)d∅
derivative of sinx - ANSWER-cosx
derivative of cosx - ANSWER--sinx
integral of sinx - ANSWER--cosx
integral of cosx - ANSWER-sinx
Sin(A) Sin(B) - ANSWER-1/2[Cos(A-B)-Cos(A+B)]
Cos(A) Cos(B) - ANSWER-1/2[Cos(A-B)+Cos(A+B)]
Sin(A) Cos(B) - ANSWER-1/2[Sin(A-B)+Sin(A+B)]
f average - ANSWER-(1/B-A)∫f(x)dx
Area - ANSWER-b
∫[f(x)-g(x)]dx
a
√(a²-x²) - ANSWER-x=aSinθ
√(x²-a²) - ANSWER-x=aSecθ
√(a²+x²) - ANSWER-x=aTanθ
Surface area - ANSWER-b
S=∫ 2π f(x) √(1+[f'(x)]²) dx
a
