AP Calculus AB Chapter 4: Integrals

AP Calculus AB Chapter 4: Integrals

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Section 1

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Mean Value Theorem (Formula)

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Cards (19)

Section 1

(19 cards)

Mean Value Theorem (Formula)

Front

Back

Fundamental Theorem of Calculus

Front

Back

∫f(g(x))g'(x)dx

Front

u=g(x), du=g'(x) ∫f(u)du = F(u) + C

Back

Average Value of a Function on an Interval

Front

If f is integrable on the closed interval [a,b], then the average value of f on the interval is...

Back

Net Change Theorem

Front

The definite integral of the rate of change of a quantity F'(x) gives the total change, or net change, in that quantity on the interval [a,b].

Back

∫x(4x²+3)⁴ dx

Front

u=4x²+3 du=8x dx =∫u⁴ du =(1/5)(1/8) u⁵ + C = (1/40)u⁵ + C =(1/40)(4x²+3)⁵ + C

Back

Informal Definition of Definite Integral

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Back

Definite Integral as the Area of a region

Front

Back

∫x²/7-x³

Front

u=7-x³ du=-3x² dx du/3= -x² dx =-∫1/3u du = -1/3 ∫1/u du =( -ln|u|/3) +C = (-ln|7-x³|)/3 +C

Back

U Substitution

Front

Using a change of variables to find a definite/indefinite integral

Back

Continuity Implies Integrability

Front

If a function is continuous on the closed interval [a,b], then it can be integrated on [a,b].

Back

If f is integrable on [a,b], then...

Front

The integral of [a,b] = the negative integral [b,a]

Back

Mean Value Theorem (Words)

Front

If f is continuous on the closed interval [a,b], then there exists a number c in the closed interval [a,b] such that...

Back

If f is integrable on the three closed intervals determined by a, b, and c, then...

Front

The integral of [a,b] = the integral of [a,c] + the integral of [b,c]

Back

∫cos³xsinx dx

Front

∫cos³xsinx dx u=cosx du=-sinx dx -∫u³ du = -∫u⁴/4 + C = -(cos⁴x)/4 + C

Back

Second Fundamental Theorem of Calculus

Front

If f is continuous on an open interval I containing a, then, for every x in the interval,....

Back

If f is defined at x=a, then...

Front

The integral from [a,a] = 0

Back

Steps to Substitute In Indefinite Integrals

Front

1) Define "u" for change of variables. Usually "u" will be the inner function in a composite function 2) Differentiate "u" to find "du" 3) Substitute in the integrand and simplify 4) If "x" still occurs anywhere in the integrand, take "u", solve for "x" in terms of "u", substitute, and simplify 5) Integrate 6) Substitute back for "u", so the answer is in terms of "x"

Back

Steps to Substitute In Definite Integrals

Front

1) Define "u" 2) Differentiate "u" to find "du" 3) Substitute in the integrand and simplify 4) Use the substitution to change the limits of integration 5) If "x" still occurs anywhere in the integrand, take "u", solve for "x" in terms of "u", substitute, and simplify 6) Integrate 7) Substitute back for "u", so the answer is in terms of "x"

Back