CALC 2 MIDTERM EXAM WITH SOLUTIONS 2025-26 #7
Area between two curves , y = f(x) - correct answer ∫ f(x) - g(x) dx, from a to b.
(top - bottom)
(outer - inner)
F(x)≥g(x)
Area between two curves, x = f(y) - correct answer ∫ f(y) - g(y) dy, from c to d.
(right - left)
(outer - inner)
F(y)≥g(y)
Washer (aka disk) method, about x axis. - correct answer π∫ f(x)² - g(x)² dx, from a to b.
If above x axis,
Subtract number from each function.
If below, add number to each function.
(top- bottom)
(outer - inner)
F(x)≥g(x)
Washer (aka disk) method, about y axis. - correct answer π∫ f(y)² - g(y)² dy, from c to d.
If larger than y axis, subtract from each function
If smaller than y axis, add to each function.
(right - left)
(outer - inner)
F(y)≥g(y)
Shell method, about x axis - correct answer 2π∫y*f(y) dy, from c to d.
If larger than x axis, subtract number from the RADIUS ONLY.
If smaller than x axis, subtract number from the RADIUS ONLY.
Shell method, about y axis - correct answer 2π∫x*f(x) dx, from a to b.
, (radius * function)
If larger than y axis, subtract number from the RADIUS ONLY
If smaller than y axis, add number to RADIUS ONLY.
Shell method for 2 functions - correct answer 2π∫(x* (f(x) - g(x))dx, from a to b.
TOP - BOTTOM
2π∫(y* (f(y) - g(y))dy, from c to d.
RIGHT - LEFT
Shell method in general - correct answer You rotate it on the axis OPPOSITE of what it
is in respect to.
For example:
If rotating about y axis,
Y = f(x).
Integration by Parts - correct answer ∫uvd = uv - ∫vdu
Arc length - correct answer ∫√(1 + f'(x)²)dx, from a to b.
∫√(1+f'(y)²)dy, from c to d.
Arcsin' u - correct answer 1/√(1-u²)
Arccos'u - correct answer -1/√(1 - u²)
Arctan' u - correct answer 1/(1 + u²)
Arccot' u - correct answer -1/(1 + x²)
√(a²-b²x²) - correct answer x = (a/b)sinθ
Cos²θ = 1-sin²θ
√(b²x²-a²) - correct answer x = (a/b)secθ
Tan²θ = sec²θ-1
√(a²+ b²x²) - correct answer x = (a/b)tanθ
Sec²θ=1+tan²θ
Area between two curves , y = f(x) - correct answer ∫ f(x) - g(x) dx, from a to b.
(top - bottom)
(outer - inner)
F(x)≥g(x)
Area between two curves, x = f(y) - correct answer ∫ f(y) - g(y) dy, from c to d.
(right - left)
(outer - inner)
F(y)≥g(y)
Washer (aka disk) method, about x axis. - correct answer π∫ f(x)² - g(x)² dx, from a to b.
If above x axis,
Subtract number from each function.
If below, add number to each function.
(top- bottom)
(outer - inner)
F(x)≥g(x)
Washer (aka disk) method, about y axis. - correct answer π∫ f(y)² - g(y)² dy, from c to d.
If larger than y axis, subtract from each function
If smaller than y axis, add to each function.
(right - left)
(outer - inner)
F(y)≥g(y)
Shell method, about x axis - correct answer 2π∫y*f(y) dy, from c to d.
If larger than x axis, subtract number from the RADIUS ONLY.
If smaller than x axis, subtract number from the RADIUS ONLY.
Shell method, about y axis - correct answer 2π∫x*f(x) dx, from a to b.
, (radius * function)
If larger than y axis, subtract number from the RADIUS ONLY
If smaller than y axis, add number to RADIUS ONLY.
Shell method for 2 functions - correct answer 2π∫(x* (f(x) - g(x))dx, from a to b.
TOP - BOTTOM
2π∫(y* (f(y) - g(y))dy, from c to d.
RIGHT - LEFT
Shell method in general - correct answer You rotate it on the axis OPPOSITE of what it
is in respect to.
For example:
If rotating about y axis,
Y = f(x).
Integration by Parts - correct answer ∫uvd = uv - ∫vdu
Arc length - correct answer ∫√(1 + f'(x)²)dx, from a to b.
∫√(1+f'(y)²)dy, from c to d.
Arcsin' u - correct answer 1/√(1-u²)
Arccos'u - correct answer -1/√(1 - u²)
Arctan' u - correct answer 1/(1 + u²)
Arccot' u - correct answer -1/(1 + x²)
√(a²-b²x²) - correct answer x = (a/b)sinθ
Cos²θ = 1-sin²θ
√(b²x²-a²) - correct answer x = (a/b)secθ
Tan²θ = sec²θ-1
√(a²+ b²x²) - correct answer x = (a/b)tanθ
Sec²θ=1+tan²θ
