MAT 271 FINAL EXAM UPDATED QUESTIONS AND
ANSWERS SURE A+
✔✔trig substitution for integrals - ✔✔1) set x equal to theta expression
---a^2 + x^2 -> x = a tan (theta)
---a^2 - x^2 -> x = a sin (theta)
2) find dx (take derivative)
3) find theta (take inverse and draw triangle)
4) replace x and dx of og integral in terms of theta
5) solve integral
---can use other trig rules to help
6) put back in terms of x using triangle
✔✔partial fraction decomposition - ✔✔-method of separating fractions
-degree of numerator MUST be less than denominator degree
---otherwise do long division to make it so
1) separate denominator
---set original fraction equal to fractions with unknown numerators
---1/(x+3)(x-1)^2 = A/x+3 + B/x+1 + C/(x+1)^2
2) multiply by og fraction denominator
3) solve by setting like terms equal
✔✔reduction formulas - ✔✔-given a formula to reduce an integral, keep using it until
there are no integrals left
✔✔improper integrals over infinite intervals - ✔✔-converges if limit exists, diverges if not
-improper BECAUSE of their infinite limit
✔✔improper integrals with an unbounded integrand - ✔✔-unbounded integrand: occurs
when integrand becomes infinite somewhere on interval of integration
-converges if limit exists, diverges if not
✔✔comparison theorem
(for interval convergence) - ✔✔-suppose f and g are cont. functions w/ 0 ≤ g(x) ≤ f(x) for
x≥a
---IF S a->inf f(x) dx is convergent, THEN so is S a->inf g(x) dx
---IF S a->inf g(x) dx is divergent, THEN so is S a-> inf f(x) dx
-pretty much squeeze theorem for interval convergence
✔✔area between curves - ✔✔-IF f(x) is bigger / "on top" and f/g are cont. and ab/b are
points of intersection
---THEN A = S ab [f(x) - g(x)] dx
---if curves recross, add them together separately
, ---if sideways (in terms of y), use dy
✔✔mean value theorem for integrals - ✔✔IF f is cont on [a,b]
THEN there is a c between a and b such that
f(c) = f bar = [1/(b-a)] [S ab f(x) dx]
-f(c) is the average value between those 2 intervals, and c is the place in which it occurs
✔✔disc / washer method - ✔✔-perpendicular to axis of rotation (use to find d__)
-takes areas of cross-sections and multiplies by width over an integral
-disc: when there is no space between volume and axis of rotation
-washer: there is a hole between volume and axis of rotation (2 functions usually)
-if axis of rotation is not an axis, get radius by axis - function
---if axis of rotation is y=3 and we are rotating y=x^3,
radius is 3 - x^3
✔✔shell method - ✔✔-parallel to axis of rotation
-finds area of thin peels of volumes over an integral
-radius: x or y, height: function
-if axis of rotation is not an axis, get radius by axis - function
---if axis of rotation is y=3 and we are rotating y=x^3,
radius is 3 - x^3
✔✔arc length (formula provided) - ✔✔-L = S ab ds
-L = S ab [ sqrt (1+ (dy/dx)^2) ] dx
-L = S ab [ sqrt (1+ (dx/dx)^2) ] dy
-L = S ab [ sqrt [ (dy/dt)^2 + (dx/dt)^2 ]] dt
✔✔centroid / center of mass - ✔✔-formulas above will be given
-A is integral of f(x) ( ab S f(x) dx )
-if applies to 2 equations, f(x) and g(x) are squared individually while x(f(x) - g(x))
-if applies to 3 lines, add 2 centroids of different intervals together to get each
coordinate
-point P on which a thin plate of any given shape balances
-x-coordinate = My / m
-y-coordinate = Mx / m
-My = n, i=1 ∑(mi xi)
-Mx = n, i=1 ∑(mi yi)
-m = n, i=1 ∑(mi)
✔✔sequence - ✔✔-{an} or {an} n=1, inf = {a1, a2, a3, ... an, ... }
-pay attention IF n=1 at beginning* given in 3 ways...
ANSWERS SURE A+
✔✔trig substitution for integrals - ✔✔1) set x equal to theta expression
---a^2 + x^2 -> x = a tan (theta)
---a^2 - x^2 -> x = a sin (theta)
2) find dx (take derivative)
3) find theta (take inverse and draw triangle)
4) replace x and dx of og integral in terms of theta
5) solve integral
---can use other trig rules to help
6) put back in terms of x using triangle
✔✔partial fraction decomposition - ✔✔-method of separating fractions
-degree of numerator MUST be less than denominator degree
---otherwise do long division to make it so
1) separate denominator
---set original fraction equal to fractions with unknown numerators
---1/(x+3)(x-1)^2 = A/x+3 + B/x+1 + C/(x+1)^2
2) multiply by og fraction denominator
3) solve by setting like terms equal
✔✔reduction formulas - ✔✔-given a formula to reduce an integral, keep using it until
there are no integrals left
✔✔improper integrals over infinite intervals - ✔✔-converges if limit exists, diverges if not
-improper BECAUSE of their infinite limit
✔✔improper integrals with an unbounded integrand - ✔✔-unbounded integrand: occurs
when integrand becomes infinite somewhere on interval of integration
-converges if limit exists, diverges if not
✔✔comparison theorem
(for interval convergence) - ✔✔-suppose f and g are cont. functions w/ 0 ≤ g(x) ≤ f(x) for
x≥a
---IF S a->inf f(x) dx is convergent, THEN so is S a->inf g(x) dx
---IF S a->inf g(x) dx is divergent, THEN so is S a-> inf f(x) dx
-pretty much squeeze theorem for interval convergence
✔✔area between curves - ✔✔-IF f(x) is bigger / "on top" and f/g are cont. and ab/b are
points of intersection
---THEN A = S ab [f(x) - g(x)] dx
---if curves recross, add them together separately
, ---if sideways (in terms of y), use dy
✔✔mean value theorem for integrals - ✔✔IF f is cont on [a,b]
THEN there is a c between a and b such that
f(c) = f bar = [1/(b-a)] [S ab f(x) dx]
-f(c) is the average value between those 2 intervals, and c is the place in which it occurs
✔✔disc / washer method - ✔✔-perpendicular to axis of rotation (use to find d__)
-takes areas of cross-sections and multiplies by width over an integral
-disc: when there is no space between volume and axis of rotation
-washer: there is a hole between volume and axis of rotation (2 functions usually)
-if axis of rotation is not an axis, get radius by axis - function
---if axis of rotation is y=3 and we are rotating y=x^3,
radius is 3 - x^3
✔✔shell method - ✔✔-parallel to axis of rotation
-finds area of thin peels of volumes over an integral
-radius: x or y, height: function
-if axis of rotation is not an axis, get radius by axis - function
---if axis of rotation is y=3 and we are rotating y=x^3,
radius is 3 - x^3
✔✔arc length (formula provided) - ✔✔-L = S ab ds
-L = S ab [ sqrt (1+ (dy/dx)^2) ] dx
-L = S ab [ sqrt (1+ (dx/dx)^2) ] dy
-L = S ab [ sqrt [ (dy/dt)^2 + (dx/dt)^2 ]] dt
✔✔centroid / center of mass - ✔✔-formulas above will be given
-A is integral of f(x) ( ab S f(x) dx )
-if applies to 2 equations, f(x) and g(x) are squared individually while x(f(x) - g(x))
-if applies to 3 lines, add 2 centroids of different intervals together to get each
coordinate
-point P on which a thin plate of any given shape balances
-x-coordinate = My / m
-y-coordinate = Mx / m
-My = n, i=1 ∑(mi xi)
-Mx = n, i=1 ∑(mi yi)
-m = n, i=1 ∑(mi)
✔✔sequence - ✔✔-{an} or {an} n=1, inf = {a1, a2, a3, ... an, ... }
-pay attention IF n=1 at beginning* given in 3 ways...
