AP Calculus AB · Flashcards
Unit 6 — Applications of Integration
Course framework: Unit 8, plus Riemann and trapezoidal sums from Unit 6 · work cards are extra (not on the AP exam) · 87 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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Rate → Amount & Numerical Sums✓ Known #1
If \(R(t)\) is a rate (units per time), what does \(\int_a^b R(t)\,dt\) represent?
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Answer#1
The net amount accumulated (net change) from \(t=a\) to \(t=b\).
Trap: Leaving the answer in rate units.
Rate → Amount & Numerical Sums✓ Known #2
If a rate can be negative, what does its integral give?
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Answer#2
The net change: positive contributions minus negative ones.
Trap: Calling it a total without checking the sign.
Rate → Amount & Numerical Sums✓ Known #3
When do you integrate \(R\) and when \(|R|\)?
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Answer#3
- Net change, or the amount at a time (initial amount + \(\int R\)): integrate \(R\).
- A total that cannot cancel, such as total distance traveled: integrate \(|R|\).
Trap: Automatically using the absolute value.
Rate → Amount & Numerical Sums✓ Known #4
If \(R(t)\) is in gallons per hour, what are the units of \(\int R(t)\,dt\)?
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Answer#4
Gallons.
Trap: Answering gallons per hour.
Rate → Amount & Numerical Sums✓ Known #5
If \(v(t)\) is velocity, what is \(\int_a^b v(t)\,dt\)?
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Answer#5
The displacement on \([a,b]\).
Trap: Calling it distance traveled.
Rate → Amount & Numerical Sums✓ Known #6
If \(r(t)\) is the rate at which people enter a store (people/min), what is \(\int_a^b r(t)\,dt\)?
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Answer#6
The number of people who enter the store from \(t=a\) to \(t=b\).
Trap: Multiplying one rate value by the time, as if the rate were constant.
Rate → Amount & Numerical Sums✓ Known #7
If \(T'(t)\) is the rate of change of temperature (°C/min), what is \(\int_a^b T'(t)\,dt\)?
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Answer#7
\(T(b)-T(a)\): the net change in temperature, in °C.
Trap: Interpreting it as “the total temperature.”
Rate → Amount & Numerical Sums✓ Known #8
Average value of a rate \(R(t)\) on \([a,b]\)?
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Answer#8
\(\dfrac{1}{b-a}\displaystyle\int_a^b R(t)\,dt\).
Trap: Using \(\frac{R(b)-R(a)}{b-a}\), which is the average rate of change of \(R\).
Rate → Amount & Numerical Sums✓ Known #9
If \(\int_0^{10} R(t)\,dt=-25\), what does the negative sign mean?
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Answer#9
The quantity decreased by a net 25 units over \(0\le t\le10\): more outflow than inflow.
Trap: Saying “the amount is negative.”
Rate → Amount & Numerical Sums✓ Known #10
AP-style sentence: interpret \(\int_a^b r(t)\,dt\) in context.
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Answer#10
“\(\int_a^b r(t)\,dt\) is the net amount of [quantity, with units] that [enters/changes] from time \(a\) to time \(b\).”
Trap: Not naming the quantity, the units or the interval.
Rate → Amount & Numerical Sums✓ Known #11
Given the graph of a rate, how do you estimate the accumulated change?
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Answer#11
Find the signed area under the rate graph over the interval.
Trap: Reading a single \(y\)-value as the total change.
Rate → Amount & Numerical Sums✓ Known #12
If the rate graph is piecewise linear, what is the quickest way to integrate?
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Answer#12
Use geometry: triangles, rectangles and trapezoids, with signs.
Trap: Forgetting regions below the axis are negative.
Rate → Amount & Numerical Sums✓ Known #13
Trapezoidal sum with equal widths?
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Answer#13
\(\displaystyle\int_a^b f \approx \Delta x\left[\tfrac12 f(x_0)+f(x_1)+\cdots+f(x_{n-1})+\tfrac12 f(x_n)\right]\).
Trap: Forgetting the half-weights on the endpoints.
Rate → Amount & Numerical Sums✓ Known #14
Midpoint sum with \(n\) subintervals?
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Answer#14
\(\displaystyle\int_a^b f\approx \Delta x\sum_{i=1}^{n} f(m_i)\), where the \(m_i\) are the midpoints.
Trap: Using endpoints instead of midpoints.
Rate → Amount & Numerical Sums✓ Known #15
If \(f\) is concave up, does a trapezoidal sum over- or underestimate?
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Answer#15
Overestimates.
Trap: Mixing it up with the midpoint sum.
Rate → Amount & Numerical Sums✓ Known #16
If \(f\) is concave up, does a midpoint sum over- or underestimate?
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Answer#16
Underestimates.
Trap: Reversing the rule.
Rate → Amount & Numerical Sums✓ Known #17
If \(f\) is concave down, does a trapezoidal sum over- or underestimate?
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Answer#17
Underestimates.
Trap: Swapping the concavity.
Rate → Amount & Numerical Sums✓ Known #18
If \(f\) is concave down, does a midpoint sum over- or underestimate?
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Answer#18
Overestimates.
Trap: Swapping midpoint and trapezoid.
Area Between Curves✓ Known #19
Area between curves (integrating in \(x\))?
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Answer#19
\(\displaystyle\int_a^b \big[\text{top}(x)-\text{bottom}(x)\big]\,dx\).
Trap: Integrating bottom minus top and getting a negative.
Area Between Curves✓ Known #20
How do you find the limits for the area between curves?
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Answer#20
Set the functions equal and solve for the intersection points.
Trap: Using the window of the graph without justification.
Area Between Curves✓ Known #21
If the top curve changes inside the interval, what must you do?
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Answer#21
Split the integral where the curves switch order.
Trap: Using one integral and letting parts cancel.
Area Between Curves✓ Known #22
Area between curves integrating in \(y\)?
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Answer#22
\(\displaystyle\int_c^d \big[\text{right}(y)-\text{left}(y)\big]\,dy\).
Trap: Using top minus bottom with \(dy\).
Area Between Curves✓ Known #23
When is integrating in \(y\) helpful?
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Answer#23
When horizontal slices give simpler expressions, or the curves are given as \(x\) in terms of \(y\).
Trap: Forgetting to change the limits to \(y\)-values.
Area Between Curves✓ Known #24
Total area between \(f\) and the \(x\)-axis on \([a,b]\)?
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Answer#24
\(\int_a^b |f(x)|\,dx\) — split where \(f=0\).
Trap: Using \(\left|\int_a^b f\right|\).
Area Between Curves✓ Known #25
If \(\int_a^b f(x)\,dx=0\), must \(f\) be zero on \([a,b]\)?
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Answer#25
No. Positive and negative areas can cancel.
Trap: Concluding \(f(x)=0\) everywhere.
Area Between Curves✓ Known #26
If the region under a line is a triangle, what is the integral?
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Answer#26
The area of the triangle, with a negative sign if it lies below the \(x\)-axis.
Trap: Forgetting the sign below the axis.
Area Between Curves✓ Known #27
If \(f\) and \(g\) intersect at \(x=a\) and \(x=b\), what is true there?
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Answer#27
\(f(a)=g(a)\) and \(f(b)=g(b)\).
Trap: Thinking the slopes must match.
Area Between Curves✓ Known #28
“Find the area of the region enclosed by …”: what should you check first?
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Answer#28
That the curves really enclose a finite region — usually two intersection points or given boundaries.
Trap: Integrating over an interval that does not enclose a region.
Area Between Curves✓ Known #29
What should a final area answer look like?
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Answer#29
A positive number, with square units if units are given.
Trap: Leaving a negative value.
Area Between Curves✓ Known #30
Integrating right minus left with \(dy\): what are the limits \(c\) and \(d\)?
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Answer#30
The \(y\)-values where the region begins and ends (usually intersection \(y\)-values).
Trap: Using \(x\)-values by mistake.
Area Between Curves✓ Known #31
If the curves are \(x=R(y)\) (right) and \(x=L(y)\) (left), what is the area?
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Answer#31
\(\displaystyle\int_c^d \big[R(y)-L(y)\big]\,dy\).
Trap: Swapping \(R\) and \(L\).
Area Between Curves✓ Known #32
If two curves cross at exactly one point, can they enclose a region by themselves?
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Answer#32
Usually not — you need another boundary (a line, an axis or another curve).
Trap: Forcing an “enclosed area” setup.
Area Between Curves✓ Known #33
“Area between the curve and the \(y\)-axis”: what is the boundary?
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Answer#33
The line \(x=0\); integrating in \(y\) (right minus left) is often easiest.
Trap: Using \(x\)-limits that do not reach the \(y\)-axis.
Area Between Curves✓ Known #34
If \(f(x)\ge g(x)\) on \([a,b]\), what is the area between them?
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Answer#34
\(\displaystyle\int_a^b \big(f(x)-g(x)\big)\,dx\).
Trap: Forgetting to split if the order changes.
Volume (Cross Sections, Disks, Washers)✓ Known #35
Volume with known cross sections perpendicular to the \(x\)-axis?
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Answer#35
\(V=\displaystyle\int_a^b A(x)\,dx\), where \(A(x)\) is the area of the cross section at \(x\).
Trap: Adding \(\pi\) when the cross sections are not circles.
Volume (Cross Sections, Disks, Washers)✓ Known #36
Cross sections are squares whose side is the region’s width \(s(x)\). What is \(A(x)\)?
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Answer#36
\(A(x)=[s(x)]^2\), so \(V=\displaystyle\int_a^b [s(x)]^2\,dx\).
Trap: Using the height of one curve instead of the width between the curves.
Volume (Cross Sections, Disks, Washers)✓ Known #37
Cross sections are semicircles with diameter \(s(x)\). What is \(A(x)\)?
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Answer#37
\(A(x)=\dfrac12\pi\left(\dfrac{s(x)}{2}\right)^2=\dfrac{\pi}{8}[s(x)]^2\).
Trap: Using the diameter as the radius.
Volume (Cross Sections, Disks, Washers)✓ Known #38
Cross sections are equilateral triangles with side \(s(x)\). What is \(A(x)\)?
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Answer#38
\(A(x)=\dfrac{\sqrt3}{4}[s(x)]^2\).
Trap: Using \(\frac12 s^2\), which is a right isosceles triangle with legs \(s\).
Volume (Cross Sections, Disks, Washers)✓ Known #39
Disk method about the \(x\)-axis with radius \(r(x)\)?
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Answer#39
\(V=\pi\displaystyle\int_a^b [r(x)]^2\,dx\).
Trap: Forgetting to square the radius.
Volume (Cross Sections, Disks, Washers)✓ Known #40
Washer method about the \(x\)-axis with outer radius \(R(x)\) and inner radius \(r(x)\)?
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Answer#40
\(V=\pi\displaystyle\int_a^b \big(R(x)^2-r(x)^2\big)\,dx\).
Trap: Using \((R-r)^2\) instead of \(R^2-r^2\).
Volume (Cross Sections, Disks, Washers)✓ Known #41
Rotating about the \(y\)-axis with disks or washers: what changes?
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Answer#41
Integrate with respect to \(y\): the radius is a horizontal distance, \(x\) written as a function of \(y\), and the limits are \(y\)-values.
Trap: Using \(y\)-values as radii, or keeping \(x\)-limits.
Volume (Cross Sections, Disks, Washers)✓ Known #42
Rotating about a horizontal line \(y=c\): how do you form the radius?
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Answer#42
Radius = vertical distance from the curve to \(y=c\).
Trap: Forgetting to subtract \(c\).
Volume (Cross Sections, Disks, Washers)✓ Known #43
Rotating about a vertical line \(x=c\) (integrating in \(y\)): how do you form the radius?
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Answer#43
Radius = horizontal distance from the curve to \(x=c\).
Trap: Using a vertical distance.
Volume (Cross Sections, Disks, Washers)✓ Known #44
The region between \(y=f(x)\ge0\) and \(y=0\) is rotated about the \(x\)-axis. Which method?
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Answer#44
Disks: \(V=\pi\displaystyle\int_a^b [f(x)]^2\,dx\).
Trap: Adding an inner radius that does not exist.
Volume (Cross Sections, Disks, Washers)✓ Known #45
The region between \(y=f(x)\) and \(y=g(x)\) (both above the axis, \(f\ge g\)) is rotated about the \(x\)-axis. What are the radii?
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Answer#45
Outer radius \(R=f(x)\), inner radius \(r=g(x)\).
Trap: Setting \(R=f-g\).
Volume (Cross Sections, Disks, Washers)✓ Known #46
If part of the region is below the \(x\)-axis and you rotate about the \(x\)-axis, what must you watch?
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Answer#46
Radii are distances to the axis; think in absolute values before squaring, and check which curve is farther.
Trap: Choosing the outer radius by “top curve” instead of distance.
Volume (Cross Sections, Disks, Washers)✓ Known #47
Is the shell method part of the AP Calculus course?
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Answer#47
No. AP uses disks, washers and known cross sections; for rotation about a vertical axis, integrate with respect to \(y\).
Trap: Spending time on shells for the AP exam.
Volume (Cross Sections, Disks, Washers)✓ Known #48
If \(x\) and \(y\) are in meters, what are the units of volume?
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Answer#48
Cubic meters (m³).
Trap: Using square units.
Volume (Cross Sections, Disks, Washers)✓ Known #49
How do you avoid forgetting \(\pi\) in disk and washer problems?
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Answer#49
Start by writing \(V=\pi\int\ldots\) before substituting the radii.
Trap: Adding \(\pi\) at the end inconsistently.
Volume (Cross Sections, Disks, Washers)✓ Known #50
Washer method about \(y=k\)?
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Answer#50
\(V=\pi\displaystyle\int_a^b \Big[(\text{outer distance to }y=k)^2-(\text{inner distance to }y=k)^2\Big]\,dx\).
Trap: Using the \(y\)-values directly instead of distances to the axis.
Volume (Cross Sections, Disks, Washers)✓ Known #51
If the region touches the axis of rotation along the whole interval, what is the inner radius?
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Answer#51
0 — use disks.
Trap: Inventing an inner radius.
Volume (Cross Sections, Disks, Washers)✓ Known #52
Rotating about \(y=2\), with the curve \(y=f(x)\) above 2: what is the radius?
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Answer#52
\(r(x)=f(x)-2\).
Trap: Using \(2-f(x)\).
Volume (Cross Sections, Disks, Washers)✓ Known #53
Rotating about \(y=2\), with the curve below 2: what is the radius?
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Answer#53
\(r(x)=2-f(x)\).
Trap: Using \(f(x)-2\), which is negative as a distance.
Volume (Cross Sections, Disks, Washers)✓ Known #54
Region between \(f\) and \(g\) rotated about \(y=c\): how do you choose the radii?
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Answer#54
\(R=\) distance from \(y=c\) to the farther curve; \(r=\) distance to the closer curve.
Trap: Choosing by top/bottom instead of distance to the axis.
Volume (Cross Sections, Disks, Washers)✓ Known #55
What are the limits \(a\) and \(b\) in a volume integral?
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Answer#55
The \(x\)-values (or \(y\)-values) over which the region exists — usually intersection points or given bounds.
Trap: Using where the axis of rotation crosses the curves.
Volume (Cross Sections, Disks, Washers)✓ Known #56
What changes when you integrate a washer problem with respect to \(y\)?
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Answer#56
The radii become functions of \(y\) and the limits become \(y\)-values; the pattern \(\pi\int(R^2-r^2)\) stays the same.
Trap: Keeping \(x\)-limits while integrating \(dy\).
Volume (Cross Sections, Disks, Washers)✓ Known #57
What should a correct volume setup show, even unevaluated?
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Answer#57
The integral with its limits, correct radius (or area) expressions, and the factor \(\pi\) when the sections are circular.
Trap: Writing “use the washer method” without an integral.
Volume (Cross Sections, Disks, Washers)✓ Known #58
Can a volume be negative?
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Answer#58
No. Volume is always nonnegative.
Trap: Reporting a negative after reversing the limits.
Volume (Cross Sections, Disks, Washers)✓ Known #59
Squared radii: does the sign of the radius expression matter?
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Answer#59
Not for the square, but the radius must still be the correct distance to the axis.
Trap: Letting a wrong radius hide behind squaring.
Volume (Cross Sections, Disks, Washers)✓ Known #60
When do you need to split a washer integral?
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Answer#60
When the outer and inner curves switch roles inside the interval.
Trap: Assuming one order holds everywhere.
Average Value✓ Known #61
Average value of \(f\) on \([a,b]\)?
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Answer#61
\(f_{\text{avg}}=\dfrac{1}{b-a}\displaystyle\int_a^b f(x)\,dx\).
Trap: Using the average-rate-of-change formula.
Average Value✓ Known #62
If \(f\) is velocity, what is its average value?
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Answer#62
The average velocity: displacement per unit time.
Trap: Calling it average speed.
Average Value✓ Known #63
What condition does the Mean Value Theorem for integrals require?
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Answer#63
\(f\) continuous on \([a,b]\).
Trap: Requiring differentiability (that is the derivative MVT).
Average Value✓ Known #64
Conclusion of the Mean Value Theorem for integrals?
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Answer#64
There is \(c\) in \([a,b]\) with \(f(c)=f_{\text{avg}}\).
Trap: Writing \(f'(c)=f_{\text{avg}}\).
Average Value✓ Known #65
How do you find \(c\) with \(f(c)=f_{\text{avg}}\)?
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Answer#65
- Compute \(f_{\text{avg}}\).
- Solve \(f(c)=f_{\text{avg}}\) for \(c\) in \([a,b]\).
Trap: Keeping solutions outside the interval.
Average Value✓ Known #66
If \(f\) is strictly increasing on \([a,b]\), where is \(c\) with \(f(c)=f_{\text{avg}}\)?
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Answer#66
In \((a,b)\), and it is unique — but not necessarily the midpoint.
Trap: Assuming \(c\) is the midpoint.
Motion with Integrals✓ Known #67
Given \(a(t)\) and \(v(0)=v_0\), find \(v(t)\).
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Answer#67
\(v(t)=v_0+\displaystyle\int_0^t a(u)\,du\).
Trap: Missing the initial condition.
Motion with Integrals✓ Known #68
Given \(v(t)\) and \(s(0)=s_0\), find \(s(t)\).
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Answer#68
\(s(t)=s_0+\displaystyle\int_0^t v(u)\,du\).
Trap: Forgetting \(s_0\).
Motion with Integrals✓ Known #69
Distance traveled on \([a,b]\) when \(v\) changes sign?
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Answer#69
Split at the zeros of \(v\) and compute \(\int_a^b |v(t)|\,dt\).
Trap: Taking the absolute value only at the end.
Motion with Integrals✓ Known #70
If \(v(t)>0\) on \([a,b]\), what is the distance traveled?
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Answer#70
\(\int_a^b v(t)\,dt\) — the same as the displacement.
Trap: Thinking you must always split.
Motion with Integrals✓ Known #71
Interpret \(\int_a^b v(t)\,dt\) in words.
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Answer#71
The net change in position (displacement) from time \(a\) to time \(b\).
Trap: Calling it total distance.
Motion with Integrals✓ Known #72
If \(v\) is in m/s and you integrate over seconds, what are the units?
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Answer#72
Meters.
Trap: Answering m/s.
Common Traps✓ Known #73
Can an area or a volume be negative?
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Answer#73
No — set up the integral so the result is nonnegative.
Trap: Reporting a negative because of swapped limits or top/bottom.
Common Traps✓ Known #74
Net change vs total: what is the main warning in rate problems?
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Answer#74
Net change lets positive and negative parts cancel; a total that cannot cancel (like distance) needs absolute values or splitting.
Trap: Answering a net change when a total distance is asked.
Common Traps✓ Known #75
Is \((R-r)^2\) ever correct in the washer method?
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Answer#75
No. Washers use \(R^2-r^2\).
Trap: Expanding the wrong expression.
Common Traps✓ Known #76
Integrating with respect to \(y\): what must change besides \(dy\)?
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Answer#76
The limits must be \(y\)-values and the curves must be written as \(x\) in terms of \(y\).
Trap: Keeping \(x\)-limits while integrating \(dy\).
Work (extra — not on AP exam)✓ Known #77
Not on the AP exam · Work done by a variable force?
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Answer#77
\(W=\displaystyle\int_a^b F(x)\,dx\).
Trap: Multiplying one force value by the distance.
Work (extra — not on AP exam)✓ Known #78
Not on the AP exam · Units of work with newtons and meters?
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Answer#78
Joules (N·m).
Trap: Answering newtons.
Work (extra — not on AP exam)✓ Known #79
Not on the AP exam · Hooke’s law for springs?
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Answer#79
\(F(x)=kx\), where \(x\) is the displacement from the natural length.
Trap: Using \(F=k/x\).
Work (extra — not on AP exam)✓ Known #80
Not on the AP exam · Work to stretch a spring from \(x=a\) to \(x=b\)?
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Answer#80
\(W=\displaystyle\int_a^b kx\,dx\).
Trap: Using \(k(b-a)\).
Work (extra — not on AP exam)✓ Known #81
Not on the AP exam · A force of 10 N stretches a spring 0.2 m. Find \(k\).
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Answer#81
\(k=\dfrac{F}{x}=\dfrac{10}{0.2}=50\) N/m.
Trap: Multiplying instead of dividing.
Work (extra — not on AP exam)✓ Known #82
Not on the AP exam · Why does variable-force work need an integral?
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Answer#82
Work adds up small pieces \(dW\approx F(x)\,dx\), and the force changes with position.
Trap: Using \(W=Fd\) with a nonconstant force.
Work (extra — not on AP exam)✓ Known #83
Not on the AP exam · Pumping a liquid: what is the idea?
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Answer#83
Slice the liquid; work = \(\int\) (weight of a slice) × (distance that slice is lifted).
Trap: Using one lift distance for every slice.
Work (extra — not on AP exam)✓ Known #84
Not on the AP exam · In pumping problems, which two quantities must you write in terms of \(x\)?
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Answer#84
- The weight of a slice (from its volume).
- The distance that slice is lifted.
Trap: Forgetting one of them.
Work (extra — not on AP exam)✓ Known #85
Not on the AP exam · What does a weight density (such as 62.4 lb/ft³) multiply?
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Answer#85
Volume, to give weight (a force).
Trap: Multiplying it by a distance.
Work (extra — not on AP exam)✓ Known #86
Not on the AP exam · What is the most common variable mistake in pumping problems?
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Answer#86
Measuring \(x\) from the bottom but writing the lift distance as if from the top (or vice versa).
Trap: Using the tank height as every slice’s lift distance.
Work (extra — not on AP exam)✓ Known #87
Not on the AP exam · What should a work integral clearly show?
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Answer#87
The limits, the force expression and the variable of integration.
Trap: Giving only a number.
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