Key takeaway: This AP calculus applications integration lesson turns definite integrals into real measurements. You will compute average value, displacement, total distance, area between curves, solid volumes and net change from a rate. Each topic uses the same idea: slice the quantity, write the size of one slice, and integrate. The notes match the AB course and include the formulas for discs and washers with short checks.
Applications of integration use definite integrals to measure real quantities: average value, displacement and distance, area between curves, volumes from cross sections, volumes of revolution by discs and washers, and net change from rates in and out. Each one adds up thin slices.
AP Calculus Applications Integration Lesson: key facts
- Average value: \(\frac{1}{b-a}\int_{a}^{b}f(x)\,dx\).
- Area between curves: integrate top minus bottom, or right minus left.
- Volume with cross sections: \(\int A(x)\,dx\); washers: \(\pi\int(R^{2}-r^{2})\,dx\).
- Net change: \(A(b)=A(a)+\int_{a}^{b}A^{\prime}(t)\,dt\).
What is the average value of a function?
The average value of a continuous function \(f\) on \([a,b]\) is the height of the rectangle on \([a,b]\) that has the same area as the region under the graph.
\[f_{\text{avg}}=\frac{1}{b-a}\int_{a}^{b} f(x)\,dx\]
The Mean Value Theorem for integrals says some \(c\) in \([a,b]\) has \(f(c)=f_{\text{avg}}\).
For \(f(x)=x^{2}\) on \([0,3]\), \(\int_{0}^{3}x^{2}\,dx=9\), so \(f_{\text{avg}}=\frac{9}{3}=3\).
How do integrals describe motion?
If \(v(t)\) is velocity, then \(\int_{a}^{b} v(t)\,dt\) is the displacement (net change in position) and \(\int_{a}^{b}|v(t)|\,dt\) is the total distance. To find distance, split the interval wherever \(v\) changes sign. Position is \(s(b)=s(a)+\int_{a}^{b}v(t)\,dt\).
How do you find the area between two curves?
Subtract the lower curve from the upper curve and integrate over the interval between intersections.
Here \(A=\int_{-1}^{2}[(x+2)-x^{2}]dx=\frac{9}{2}\). If the region is easier to describe with \(y\), use \(\int_{c}^{d}[x_{\text{right}}-x_{\text{left}}]dy\). When the curves cross more than twice, find every intersection and integrate each piece with its own top and bottom.
How do cross sections give volume?
Slice a solid perpendicular to an axis. If each slice has area \(A(x)\), the volume is \(V=\int_{a}^{b}A(x)\,dx\). Let \(s\) be the length of the slice in the base.
| Cross section | Area in terms of \(s\) |
|---|---|
| Square | \(s^{2}\) |
| Semicircle (diameter \(s\)) | \(\frac{\pi}{8}s^{2}\) |
| Isosceles right triangle, hypotenuse \(s\) | \(\frac{1}{4}s^{2}\) |
| Equilateral triangle | \(\frac{\sqrt{3}}{4}s^{2}\) |
How do discs and washers work?
Revolving a region about a line makes circular slices. With no hole use \(\pi R^{2}\); with a hole use \(\pi(R^{2}-r^{2})\), where \(R\) and \(r\) are measured from the axis of rotation.
- Sketch the region and the axis; draw one slice.
- Write outer radius \(R\) and inner radius \(r\) as distances from the axis.
- Integrate \(\pi(R^{2}-r^{2})\) over the limits.
Example. Revolve the region between \(y=x\) and \(y=x^{2}\) about the \(x\)-axis: \(V=\pi\int_{0}^{1}(x^{2}-x^{4})dx=\pi(\frac{1}{3}-\frac{1}{5})=\frac{2\pi}{15}\).
How does accumulation work with rates in and out?
If \(A^{\prime}(t)\) is the rate in minus the rate out, then \(A(b)=A(a)+\int_{a}^{b}A^{\prime}(t)\,dt\). The amount is largest or smallest where the rates are equal and the sign of \(A^{\prime}\) changes.
Applications of Integration: frequently asked questions
How do you find the average value of a function on an interval?
Integrate the function over \([a,b]\) and divide by the interval length \(b-a\). For \(f(x)=x^{2}\) on \([0,3]\), the integral is 9, so the average value is 3.
What is the difference between displacement and total distance?
Displacement is \(\int v(t)\,dt\), which lets backward motion cancel forward motion. Total distance is \(\int|v(t)|\,dt\), found by splitting the interval where \(v\) changes sign.
When should you integrate with respect to y to find area?
Use \(y\) when the curves are written as \(x\) in terms of \(y\), or when one \(dx\) integral would need several pieces but one \(dy\) integral needs only a right curve minus a left curve.
How do you find volume with known cross sections?
Write the area of one slice as a function of the position, using the slice length in the base as \(s\). Then integrate that area between the limits of the base.
What is the difference between the disc and washer methods?
Discs have no hole, so each slice has area \(\pi R^{2}\). Washers have a hole, so each slice has area \(\pi(R^{2}-r^{2})\). Measure both radii from the axis of rotation, even when it is not an axis.
What is the average value of a function?
Divide the integral by the interval length. For \(f(x)=6x\) on \([2,5]\), the integral is 63 and the length is 3, so the average value is 21. Notice that this equals the function value at the midpoint, because the graph is a line.
How do you tell displacement from total distance?
Displacement integrates velocity, so negative parts subtract. Total distance integrates \(|v|\). For \(v(t)=2t-6\) on \([0,6]\), the displacement is 0, yet the total distance is 18, because the particle backs up for 3 seconds and then returns. Therefore always split at the zeros of velocity.
How does the AP calculus applications integration lesson find area between curves?
Subtract bottom from top. For \(y=2x\) and \(y=x^2\), the curves meet at 0 and 2, so the area is \(\int_0^2(2x-x^2)\,dx=\frac{4}{3}\). When the boundaries are easier as \(x\) in terms of \(y\), integrate right minus left instead.
When do you use discs, washers or cross sections?
Spin the region about the \(x\)-axis and slice perpendicular to it. If a gap appears between the region and the axis, you need washers with \(\pi(R^2-r^2)\). For \(y=x^2\) on \([0,2]\), the disc volume is \(\frac{32\pi}{5}\). Known cross sections, such as squares, use the area formula instead of \(\pi r^2\).
Which habit makes integration applications easier?
Draw the region before any algebra, and label one typical slice with its length and width. Next, state the limits from the intersection points. Finally, write the units, because AP graders look for them in contextual answers.
See every chapter on the AP Calculus AB math page.
