AP Calculus AB · Flashcards
Unit 4 — Integration & Accumulation
Course framework: Unit 6, plus motion and area applications from Unit 8 · 86 cards. Answer each card in your head, flip it, then mark it Got it or Review again. Your progress is saved in this browser.
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Riemann Sums & Integral Meaning✓ Known #1
What does \(\int_a^b f(x)\,dx\) mean?
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Answer#1
The net accumulation of \(f\) from \(a\) to \(b\) — geometrically, the signed area between \(y=f(x)\) and the \(x\)-axis.
Trap: Calling it “area” without noting that area below the axis counts as negative.
Riemann Sums & Integral Meaning✓ Known #2
When does \(\int_a^b f(x)\,dx\) equal the geometric area?
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Answer#2
When \(f(x)\ge 0\) on \([a,b]\).
Trap: Ignoring the parts where \(f(x)<0\).
Riemann Sums & Integral Meaning✓ Known #3
For \(n\) equal subintervals of \([a,b]\), what is \(\Delta x\)?
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Answer#3
\(\Delta x=\dfrac{b-a}{n}\).
Trap: Using \(\frac{a-b}{n}\).
Riemann Sums & Integral Meaning✓ Known #4
Left Riemann sum in sigma form?
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Answer#4
\(\displaystyle\sum_{i=1}^{n} f(x_{i-1})\,\Delta x\), where \(x_i=a+i\Delta x\).
Trap: Using \(x_i\) instead of \(x_{i-1}\) for left endpoints.
Riemann Sums & Integral Meaning✓ Known #5
Right Riemann sum in sigma form?
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Answer#5
\(\displaystyle\sum_{i=1}^{n} f(x_i)\,\Delta x\).
Trap: Index-shift errors.
Riemann Sums & Integral Meaning✓ Known #6
Midpoint Riemann sum in sigma form?
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Answer#6
\(\displaystyle\sum_{i=1}^{n} f\!\left(\tfrac{x_{i-1}+x_i}{2}\right)\Delta x\).
Trap: Using an endpoint instead of the midpoint.
Riemann Sums & Integral Meaning✓ Known #7
Write the \(i\)-th midpoint in terms of \(a\), \(\Delta x\) and \(i\).
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Answer#7
\(x_i^*=a+\left(i-\tfrac12\right)\Delta x\).
Trap: Using \(a+i\Delta x\), which is the right endpoint.
Riemann Sums & Integral Meaning✓ Known #8
Trapezoidal sum with equal widths: what is the formula?
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Answer#8
\(T=\dfrac{\Delta x}{2}\big[f(x_0)+2f(x_1)+\cdots+2f(x_{n-1})+f(x_n)\big]\). With unequal widths, add the trapezoids \(\frac{f(x_{i-1})+f(x_i)}{2}\,\Delta x_i\) one by one.
Trap: Doubling the first and last values, or using one \(\Delta x\) when the table spacing is unequal.
Riemann Sums & Integral Meaning✓ Known #9
If \(f\) is concave up, is a trapezoidal sum an over- or underestimate?
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Answer#9
An overestimate (the chords lie above the graph). If \(f\) is concave down, it is an underestimate.
Trap: Using increasing/decreasing instead of concavity for trapezoids.
Riemann Sums & Integral Meaning✓ Known #10
Write \(\int_a^b f(x)\,dx\) as a limit of Riemann sums.
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Answer#10
\(\displaystyle\int_a^b f(x)\,dx=\lim_{n\to\infty}\sum_{i=1}^{n} f(x_i)\,\Delta x\), with \(\Delta x=\frac{b-a}{n}\) and \(x_i=a+i\Delta x\).
Trap: Leaving out \(\Delta x\) or the limit.
Riemann Sums & Integral Meaning✓ Known #11
Which integral equals \(\displaystyle\lim_{n\to\infty}\sum_{i=1}^{n}\left(\frac{i}{n}\right)^2\frac{1}{n}\)?
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Answer#11
- \(\Delta x=\frac1n\) and \(x_i=\frac in\), so the interval is \([0,1]\).
- The limit is \(\displaystyle\int_0^1 x^2\,dx=\frac13\).
Trap: Misreading \(\Delta x\) and getting the wrong interval.
Riemann Sums & Integral Meaning✓ Known #12
If \(f\) is a rate (units per time), what are the units of \(\int f\,dt\)?
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Answer#12
(units of \(f\)) × (units of \(t\)). Example: (m/s) × s = m.
Trap: Keeping the rate units.
Riemann Sums & Integral Meaning✓ Known #13
If \(f\) is positive and decreasing, how do left and right sums compare with the integral?
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Answer#13
- Left sum overestimates.
- Right sum underestimates.
Trap: Mixing up increasing and decreasing.
Riemann Sums & Integral Meaning✓ Known #14
If \(f\) is positive and increasing, how do left and right sums compare with the integral?
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Answer#14
- Left sum underestimates.
- Right sum overestimates.
Trap: Swapping the conclusion.
Riemann Sums & Integral Meaning✓ Known #15
What is \(\int_a^b f\,dx+\int_b^c f\,dx\)?
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Answer#15
\(\int_a^c f(x)\,dx\).
Trap: Forgetting orientation when the limits are reversed.
Riemann Sums & Integral Meaning✓ Known #16
What is \(\int_b^a f(x)\,dx\)?
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Answer#16
\(-\int_a^b f(x)\,dx\).
Trap: Dropping the negative.
Riemann Sums & Integral Meaning✓ Known #17
What is \(\int_a^b c\,f(x)\,dx\)?
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Answer#17
\(c\int_a^b f(x)\,dx\).
Trap: Applying the constant to only part of the integrand.
Riemann Sums & Integral Meaning✓ Known #18
What is \(\int_a^b [f(x)+g(x)]\,dx\)?
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Answer#18
\(\int_a^b f(x)\,dx+\int_a^b g(x)\,dx\).
Trap: Thinking a similar rule works for products.
Riemann Sums & Integral Meaning✓ Known #19
If \(f\) is odd, what is \(\int_{-a}^{a} f(x)\,dx\)?
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Answer#19
\(0\).
Trap: Using the even-function rule.
Riemann Sums & Integral Meaning✓ Known #20
If \(f\) is even, what is \(\int_{-a}^{a} f(x)\,dx\)?
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Answer#20
\(2\int_0^a f(x)\,dx\).
Trap: Forgetting the factor 2.
Riemann Sums & Integral Meaning✓ Known #21
On a graph, how does a region below the \(x\)-axis affect the definite integral?
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Answer#21
It contributes a negative value (signed area).
Trap: Adding it as positive area.
Riemann Sums & Integral Meaning✓ Known #22
How do you compute the total area between a curve and the \(x\)-axis on \([a,b]\)?
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Answer#22
\(\int_a^b |f(x)|\,dx\) — split at the zeros of \(f\).
Trap: Taking \(\left|\int_a^b f(x)\,dx\right|\), which is different when \(f\) changes sign.
Accumulation (FTC Part 1)✓ Known #23
Net change: if \(F’=f\), what is \(\int_a^b f(x)\,dx\)?
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Answer#23
\(F(b)-F(a)\): the integral of a rate of change is the net change.
Trap: Confusing this with FTC Part 1.
Accumulation (FTC Part 1)✓ Known #24
If \(A(x)=\int_a^x f(t)\,dt\), what does \(A(x)\) represent?
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Answer#24
The net accumulation (signed area) under \(f\) from \(t=a\) to \(t=x\).
Trap: Calling it “the area” when \(f\) can be negative.
Accumulation (FTC Part 1)✓ Known #25
If \(A(x)=\int_a^x f(t)\,dt\), what is \(A(a)\)?
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Answer#25
\(0\).
Trap: Writing \(A(a)=f(a)\).
Accumulation (FTC Part 1)✓ Known #26
If \(A(x)=\int_a^x f(t)\,dt\), what is \(A'(x)\)?
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Answer#26
\(A'(x)=f(x)\) (FTC Part 1, for continuous \(f\)).
Trap: Writing \(A'(x)=\int f\).
Accumulation (FTC Part 1)✓ Known #27
If \(A(x)=\int_a^x f(t)\,dt\) and \(f(x)>0\), what can you say about \(A\)?
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Answer#27
\(A\) is increasing where \(f(x)>0\).
Trap: Confusing this with concavity.
Accumulation (FTC Part 1)✓ Known #28
If \(A'(x)=f(x)\) and \(f(x)<0\), what happens to \(A\)?
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Answer#28
\(A\) is decreasing.
Trap: Thinking an accumulation function can never decrease.
Accumulation (FTC Part 1)✓ Known #29
If \(A(x)=\int_a^x f(t)\,dt\), what is \(A”(x)\)?
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Answer#29
\(A”(x)=f'(x)\), where \(f\) is differentiable.
Trap: Saying \(A”=f\).
Accumulation (FTC Part 1)✓ Known #30
From a graph of \(f\), where does \(A(x)=\int_a^x f(t)\,dt\) have a local maximum?
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Answer#30
Where \(f(x)=0\) and \(f\) changes from positive to negative.
Trap: Using the slope of \(f\) instead of the sign of \(f\).
Accumulation (FTC Part 1)✓ Known #31
Where is \(A\) concave up if \(A'(x)=f(x)\)?
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Answer#31
Where \(A”(x)=f'(x)>0\), that is, where \(f\) is increasing.
Trap: Using the sign of \(f\) instead of \(f’\).
Accumulation (FTC Part 1)✓ Known #32
If \(A(x)=\int_a^x f(t)\,dt\), when does \(A\) have a horizontal tangent?
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Answer#32
When \(A'(x)=f(x)=0\).
Trap: Setting \(f'(x)=0\) instead.
Accumulation (FTC Part 1)✓ Known #33
If \(A(x)=\int_x^a f(t)\,dt\) (variable lower limit), what is \(A'(x)\)?
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Answer#33
\(A'(x)=-f(x)\).
Trap: Missing the negative sign.
Accumulation (FTC Part 1)✓ Known #34
What is \(\dfrac{d}{dx}\displaystyle\int_{g(x)}^{h(x)} f(t)\,dt\)?
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Answer#34
\(f(h(x))\,h'(x)-f(g(x))\,g'(x)\).
Trap: Forgetting to multiply by the derivatives of the limits.
Accumulation (FTC Part 1)✓ Known #35
Differentiate \(\dfrac{d}{dx}\displaystyle\int_0^{x^2} f(t)\,dt\).
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Answer#35
\(f(x^2)\cdot 2x\).
Trap: Writing only \(f(x^2)\).
Accumulation (FTC Part 1)✓ Known #36
Differentiate \(\dfrac{d}{dx}\displaystyle\int_{2x}^{5} f(t)\,dt\).
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Answer#36
\(-f(2x)\cdot 2=-2f(2x)\).
Trap: A missing sign or a missing chain factor.
FTC Part 2 & Antiderivatives✓ Known #37
FTC Part 2: if \(F’=f\), what is \(\int_a^b f(x)\,dx\)?
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Answer#37
\(F(b)-F(a)\).
Trap: Using \(F(a)-F(b)\).
FTC Part 2 & Antiderivatives✓ Known #38
\(\int x^n\,dx\) for \(n\ne -1\)?
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Answer#38
\(\dfrac{x^{n+1}}{n+1}+C\).
Trap: Forgetting \(+C\) on an indefinite integral.
FTC Part 2 & Antiderivatives✓ Known #39
\(\int \dfrac{1}{x}\,dx\)?
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Answer#39
\(\ln|x|+C\).
Trap: Dropping the absolute value.
FTC Part 2 & Antiderivatives✓ Known #40
\(\int e^x\,dx\)?
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Answer#40
\(e^x+C\).
Trap: Using the power rule on \(e^x\).
FTC Part 2 & Antiderivatives✓ Known #41
\(\int a^x\,dx\) \((a>0,\ a\ne 1)\)?
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Answer#41
\(\dfrac{a^x}{\ln a}+C\).
Trap: Multiplying by \(\ln a\) instead of dividing.
FTC Part 2 & Antiderivatives✓ Known #42
\(\int \cos x\,dx\)?
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Answer#42
\(\sin x + C\).
Trap: A sign error.
FTC Part 2 & Antiderivatives✓ Known #43
\(\int \sin x\,dx\)?
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Answer#43
\(-\cos x + C\).
Trap: Forgetting the negative.
FTC Part 2 & Antiderivatives✓ Known #44
\(\int \sec^2 x\,dx\)?
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Answer#44
\(\tan x + C\).
Trap: Writing \(\sec x\tan x\).
FTC Part 2 & Antiderivatives✓ Known #45
\(\displaystyle\int \frac{1}{1+x^2}\,dx\)?
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Answer#45
\(\arctan x + C\).
Trap: Writing \(\ln(1+x^2)\), which is the antiderivative of \(\frac{2x}{1+x^2}\).
FTC Part 2 & Antiderivatives✓ Known #46
Why is there no \(+C\) in \(\int_a^b f(x)\,dx\)?
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Answer#46
The constant cancels in \(F(b)-F(a)\).
Trap: Adding \(+C\) to a definite-integral answer.
FTC Part 2 & Antiderivatives✓ Known #47
Evaluate \(\int_a^a f(x)\,dx\).
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Answer#47
\(0\).
Trap: Saying \(f(a)\).
FTC Part 2 & Antiderivatives✓ Known #48
If \(\int_0^5 f(x)\,dx=12\), what is \(\int_5^0 f(x)\,dx\)?
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Answer#48
\(-12\).
Trap: Answering 12.
FTC Part 2 & Antiderivatives✓ Known #49
If \(G(x)=\int_1^x (t^2+1)\,dt\), find \(G'(x)\).
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Answer#49
\(G'(x)=x^2+1\).
Trap: Integrating again instead of differentiating.
FTC Part 2 & Antiderivatives✓ Known #50
If \(G(x)=\int_1^{x} (t^2+1)\,dt\), find \(G(3)\).
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Answer#50
\(\left[\frac{t^3}{3}+t\right]_1^3=(9+3)-\left(\frac13+1\right)=\frac{32}{3}\).
Trap: Forgetting to subtract the value at 1.
Average Value✓ Known #51
Average value of \(f\) on \([a,b]\)?
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Answer#51
\(f_{\text{avg}}=\dfrac{1}{b-a}\displaystyle\int_a^b f(x)\,dx\).
Trap: Forgetting the \(\frac{1}{b-a}\).
Average Value✓ Known #52
State the Mean Value Theorem for integrals.
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Answer#52
If \(f\) is continuous on \([a,b]\), there is \(c\in[a,b]\) with \(f(c)=f_{\text{avg}}\).
Trap: Confusing it with the derivative MVT.
Average Value✓ Known #53
Average value vs average rate of change?
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Answer#53
- Average value: \(\frac{1}{b-a}\int_a^b f(x)\,dx\).
- Average rate of change: \(\frac{f(b)-f(a)}{b-a}\).
Trap: Using one formula for the other.
Average Value✓ Known #54
If \(f\ge 0\), interpret \(\frac{1}{b-a}\int_a^b f(x)\,dx\) geometrically.
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Answer#54
The height of the rectangle on \([a,b]\) with the same area as the region under \(f\).
Trap: Forgetting it is an average height.
Average Value✓ Known #55
If \(v(t)\) is velocity, what is the average velocity on \([a,b]\)?
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Answer#55
\(\dfrac{1}{b-a}\displaystyle\int_a^b v(t)\,dt\): displacement divided by time.
Trap: Using \(\int|v|\), which gives average speed.
Average Value✓ Known #56
How do you find average speed on \([a,b]\)?
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Answer#56
\(\dfrac{1}{b-a}\displaystyle\int_a^b |v(t)|\,dt\) — split at the zeros of \(v\).
Trap: Using the average-velocity formula.
Average Value✓ Known #57
If \(f\) takes negative values, can \(f_{\text{avg}}\) be negative?
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Answer#57
Yes — it comes from the net signed area.
Trap: Assuming an average must be positive.
Average Value✓ Known #58
How do you find \(c\) with \(f(c)=f_{\text{avg}}\)?
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Answer#58
- Compute \(f_{\text{avg}}\).
- Solve \(f(c)=f_{\text{avg}}\) for \(c\) in \([a,b]\).
Trap: Setting \(f'(c)=f_{\text{avg}}\).
Motion with Integrals✓ Known #59
If \(v(t)\) is velocity, what is the displacement from \(a\) to \(b\)?
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Answer#59
\(\int_a^b v(t)\,dt\).
Trap: Using \(|v|\) when displacement is asked.
Motion with Integrals✓ Known #60
What is the distance traveled from \(a\) to \(b\)?
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Answer#60
\(\int_a^b |v(t)|\,dt\) — split where \(v=0\).
Trap: Taking \(\left|\int v\,dt\right|\).
Motion with Integrals✓ Known #61
If \(a(t)\) is acceleration, what is the change in velocity from \(a\) to \(b\)?
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Answer#61
\(\int_a^b a(t)\,dt=v(b)-v(a)\).
Trap: Calling it displacement.
Motion with Integrals✓ Known #62
Given \(a(t)\) and \(v(0)=v_0\), how do you find \(v(t)\)?
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Answer#62
\(v(t)=v_0+\int_0^t a(u)\,du\).
Trap: Forgetting the initial condition.
Motion with Integrals✓ Known #63
Given \(v(t)\) and \(s(0)=s_0\), how do you find \(s(t)\)?
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Answer#63
\(s(t)=s_0+\int_0^t v(u)\,du\).
Trap: Integrating \(a(t)\) instead of \(v(t)\).
Motion with Integrals✓ Known #64
If \(v(t)\) crosses 0, how does that affect total distance?
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Answer#64
Split the integral at each zero, because the sign of \(v\) changes there.
Trap: Using one integral without splitting.
Motion with Integrals✓ Known #65
If \(a(t)\) is in m/s², what are the units of \(\int a(t)\,dt\)?
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Answer#65
(m/s²) · s = m/s — a change in velocity.
Trap: Answering meters.
Motion with Integrals✓ Known #66
Average acceleration on \([a,b]\)?
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Answer#66
\(\dfrac{v(b)-v(a)}{b-a}=\dfrac{1}{b-a}\displaystyle\int_a^b a(t)\,dt\).
Trap: Using the displacement formula.
Motion with Integrals✓ Known #67
If \(s(t)\) is position, what is \(s(b)-s(a)\) called?
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Answer#67
Displacement.
Trap: Calling it distance traveled.
Motion with Integrals✓ Known #68
What is the typical sign trap in displacement problems?
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Answer#68
Negative velocity contributes negative displacement.
Trap: Treating all motion as positive distance.
Area Between Curves✓ Known #69
Area between two curves (in \(x\))?
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Answer#69
\(\displaystyle\int_a^b \big[\text{top}(x)-\text{bottom}(x)\big]\,dx\).
Trap: Reversing top and bottom and getting a negative area.
Area Between Curves✓ Known #70
How do you find the limits \(a\) and \(b\) for the area between curves?
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Answer#70
Set the functions equal and solve for the intersection points (unless the bounds are given).
Trap: Using a given interval without checking for crossings.
Area Between Curves✓ Known #71
If the curves cross inside the interval, what must you do?
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Answer#71
Split the integral where top and bottom switch.
Trap: Using one integral and letting parts cancel.
Area Between Curves✓ Known #72
When should you integrate with respect to \(y\)?
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Answer#72
When the region is easier to describe as right-minus-left curves.
Trap: Mixing \(x\)- and \(y\)-bounds.
Area Between Curves✓ Known #73
Area formula with respect to \(y\)?
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Answer#73
\(\displaystyle\int_c^d \big[\text{right}(y)-\text{left}(y)\big]\,dy\).
Trap: Using top minus bottom with \(dy\).
Area Between Curves✓ Known #74
Area bounded by a curve and the \(x\)-axis: what determines the limits?
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Answer#74
The zeros of the function and any stated boundaries.
Trap: Guessing the bounds.
Area Between Curves✓ Known #75
Area above the axis minus area below the axis gives which quantity?
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Answer#75
The net area — the definite integral.
Trap: Calling it the total area.
Area Between Curves✓ Known #76
Which integral gives the total area under \(f\) on \([a,b]\)?
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Answer#76
\(\int_a^b |f(x)|\,dx\) (split at the zeros).
Trap: Using \(\left|\int_a^b f\right|\).
u-Substitution✓ Known #77
When does \(u\)-substitution work best?
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Answer#77
When the integrand contains an inside function and (a constant multiple of) its derivative.
Trap: Forcing \(u\)-substitution without that pattern.
u-Substitution✓ Known #78
For \(\int 2x(x^2+1)^5\,dx\), choose \(u\) and find \(du\).
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Answer#78
- \(u=x^2+1\), \(du=2x\,dx\).
- Then \(\int u^5\,du=\frac{u^6}{6}+C=\frac{(x^2+1)^6}{6}+C\).
Trap: Choosing \(u=2x\) or \(u=(x^2+1)^5\).
u-Substitution✓ Known #79
Definite integrals with \(u\)-substitution: what must you do with the limits?
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Answer#79
Change the \(x\)-limits to \(u\)-limits, or substitute back to \(x\) before evaluating.
Trap: Changing the variable but keeping the old limits.
u-Substitution✓ Known #80
How do you avoid ending up with a mix of \(x\) and \(u\)?
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Answer#80
Write \(du\) explicitly and replace the matching factor of the integrand (adjusting constants).
Trap: Leaving some \(x\)’s in a \(u\)-integral.
u-Substitution✓ Known #81
\(u\)-substitution reverses which derivative rule?
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Answer#81
The chain rule.
Trap: Calling it the reverse of the product rule.
u-Substitution✓ Known #82
If you see \(\dfrac{f'(x)}{f(x)}\), what integral does it suggest?
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Answer#82
\(\displaystyle\int \frac{f'(x)}{f(x)}\,dx=\ln|f(x)|+C\).
Trap: Forgetting the absolute value.
Common Traps✓ Known #83
When does \(\left|\int_a^b f(x)\,dx\right|\) equal \(\int_a^b |f(x)|\,dx\)?
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Answer#83
Only when \(f\) does not change sign on \([a,b]\).
Trap: Assuming they are always equal.
Common Traps✓ Known #84
A rate can be negative. When do you integrate \(|\text{rate}|\) instead of the rate?
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Answer#84
- Only for totals that cannot cancel, such as total distance traveled.
- Amount at time \(t\) = initial amount + \(\int\) rate (net change).
Trap: Using \(|\text{rate}|\) to find the amount of water in a tank.
Common Traps✓ Known #85
If \(\int_a^b f(x)\,dx\) is negative, what does that mean geometrically?
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Answer#85
More area lies below the \(x\)-axis than above: the net signed area is negative.
Trap: Saying “the area is negative.” Area is never negative; signed area can be.
Common Traps✓ Known #86
If \(f\) is in gallons per hour and you integrate over hours, what are the units?
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Answer#86
Gallons.
Trap: Leaving “gallons per hour” as the final unit.
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