Applications of the Definite Integral
On a road trip across Texas, Aaliyah's family loses the odometer reading, but the car's computer still logs the speed at every moment. Can you recover how far they drove from the speed alone? You already know how to compute a definite integral. Now you will use it for real problems: areas, distance, accumulated quantities, unknown parameters, and volumes.
Net change and the Fundamental Theorem
The Fundamental Theorem of Calculus (FTC) turns integrals into subtraction. If $F'(x) = f(x)$ and $f$ is continuous on $[a, b]$, then
$$\int_a^b f(x)\,dx = F(b) - F(a) = \Big[F(x)\Big]_a^b$$Why does it work? $F'$ is the rate at which $F$ changes. Adding up all the tiny changes $F'(x)\,dx$ from $a$ to $b$ gives the total change $F(b) - F(a)$. This is the Net Change Theorem. The $+C$ never matters, because it cancels in the subtraction.
Here is the toolbox of antiderivatives you will use all page long.
| Function | Antiderivative |
|---|---|
| $x^n \ (n \ne -1)$ | $\dfrac{x^{n+1}}{n+1}$ |
| $\dfrac{1}{x}$ | $\ln\lvert x\rvert$ |
| $e^x$ | $e^x$ |
| $\sin x$ | $-\cos x$ |
| $\cos x$ | $\sin x$ |
| $\dfrac{1}{1+x^2}$ | $\arctan x$ |
Calculate the value of the following definite integral applying the Fundamental Theorem of Calculus: $$\int_{1}^{2} (3x^2 - 4x + 5) \, dx$$ <br>
Calculate the value of the following definite integral applying the Fundamental Theorem of Calculus:
$$\int_{1}^{2} (3x^2 - 4x + 5) \, dx$$The speed of a remote-controlled car (in $m/s$) is given by the function $v(t) = 2t + 1$. Determine the total distance traveled by the car between the times $t = 1$ s…
The speed of a remote-controlled car (in $m/s$) is given by the function $v(t) = 2t + 1$. Determine the total distance traveled by the car between the times $t = 1$ s and $t = 4$ s by calculating the definite integral of the speed.
An electronic components company has determined that the marginal cost of producing $x$ units of a sensor is $C'(x) = 3x^2 - 12x + 15$ dollars per unit. Calculate the…
An electronic components company has determined that the marginal cost of producing $x$ units of a sensor is $C'(x) = 3x^2 - 12x + 15$ dollars per unit. Calculate the total increase in cost when increasing production from $10$ to $20$ sensors.
Displacement versus total distance
Suppose a particle moves along a line with velocity $v(t)$. Velocity can be negative (moving backward), so there are two different questions:
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Displacement (net change in position): $\displaystyle\int_a^b v(t)\,dt$. Backward motion cancels forward motion.
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Total distance traveled: $\displaystyle\int_a^b |v(t)|\,dt$. All motion counts as positive.
To compute the second one, find where $v(t) = 0$ in the interval, check the sign of $v$ on each piece, and integrate piece by piece. On the pieces where $v < 0$, put a minus sign in front.
An autonomous vehicle moves in a straight line and its speed (in ft/s) is given by the function $v(t) = t^2 - 6t + 8$, where $t$ is the time in seconds. Calculate the…
An autonomous vehicle moves in a straight line and its speed (in ft/s) is given by the function $v(t) = t^2 - 6t + 8$, where $t$ is the time in seconds. Calculate the total distance traveled by the vehicle between the time $t = 0$ and $t = 5$. Note: Bear in mind that distance is the integral of the absolute value of the speed.
A surveillance drone moves in a straight line. Its speed in feet per second is given by the absolute value function: $v(t) = |t^2 - 6t + 5|$ for $t \in [0, 6]$ seconds.…
A surveillance drone moves in a straight line. Its speed in feet per second is given by the absolute value function:
$v(t) = |t^2 - 6t + 5|$ for $t \in [0, 6]$ seconds.
- a) Determine the total distance traveled by the drone during the first 6 seconds.
- b) Calculate the value of the time $T \in (0, 6)$ such that the distance traveled in the interval $[0, T]$ is exactly half of the total distance calculated in the previous section.
Area between a curve and the x-axis
The area between $y = f(x)$ and the x-axis on $[a, b]$ is
$$\text{Area} = \int_a^b |f(x)|\,dx$$The integral $\int_a^b f(x)\,dx$ is a signed area: regions above the axis count positive and regions below count negative. Area is never negative, so you flip the sign of the pieces below the axis.
Procedure:
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Sketch the graph, or find the zeros $f(x) = 0$ to see where it crosses the axis.
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Split $[a, b]$ at those zeros. If the problem gives no limits, the zeros are the limits.
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Integrate each piece and take its absolute value. Then add.
A graphic designer is tracing the outline of a logo that follows the curve $f(x) = e^x$. Calculate the area enclosed under this curve, the $OX$-axis and the vertical…
A graphic designer is tracing the outline of a logo that follows the curve $f(x) = e^x$. Calculate the area enclosed under this curve, the $OX$-axis and the vertical lines $x = 0$ and $x = \ln(3)$.
Calculate the area of the plane region bounded by the parabola $f(x) = x^2 - 4x + 3$ and the x-axis ($OX$). Note: Remember to first find the points of intersection of…
Calculate the area of the plane region bounded by the parabola $f(x) = x^2 - 4x + 3$ and the x-axis ($OX$). Note: Remember to first find the points of intersection of the function with the axis.
A design studio is projecting the logo for a new technology company. The base of the logo is bounded by the function $f(x) = x^3 - 4x$ and the x-axis $OX$. Calculate the…
A design studio is projecting the logo for a new technology company. The base of the logo is bounded by the function $f(x) = x^3 - 4x$ and the x-axis $OX$. Calculate the total area of the two closed regions that the curve forms with the axis.
Area between two curves
When a region lies between two graphs, slice it into thin vertical strips. At each $x$, the strip's height is top minus bottom.
Find the area of the region bounded between the graphs of the functions $f(x) = 6 - x^2$ and $g(x) = 2$. <br>
Find the area of the region bounded between the graphs of the functions $f(x) = 6 - x^2$ and $g(x) = 2$.
Calculate the area of the bounded plane region limited by the graphs of the functions $f(x) = -x^2 + 5x$ and $g(x) = x + 2$. Graphically represent the region before…
Calculate the area of the bounded plane region limited by the graphs of the functions $f(x) = -x^2 + 5x$ and $g(x) = x + 2$. Graphically represent the region before performing the calculation.
Consider the region bounded by the curves $y = e^x$, $y = e^{-x}$ and the vertical line $x = \ln(2)$. a) Draw a sketch of the region. b) Calculate the area of this…
Consider the region bounded by the curves $y = e^x$, $y = e^{-x}$ and the vertical line $x = \ln(2)$.
- a) Draw a sketch of the region.
- b) Calculate the area of this region.
When the boundaries change
Sketch first. Many regions need one of three tricks.
Trick 1: the top (or bottom) curve changes. Split the integral where the boundary changes. If the curves cross inside the interval, split at the crossing and keep top minus bottom on each piece.
Trick 2: the region is bounded by a tangent line. First find the tangent line with $y - f(x_0) = f'(x_0)(x - x_0)$. Then treat it like any other curve.
Trick 3: integrate with respect to $y$. If vertical strips would need several pieces, use horizontal strips. Solve each boundary for $x$, and the width of a strip is right minus left:
$$\text{Area} = \int_c^d \big[x_{\text{right}}(y) - x_{\text{left}}(y)\big]\,dy$$Find the area of the region bounded by the curve $f(x) = x^2 - 4x + 5$, the tangent line to this curve at the point $(3, f(3))$ and the y-axis $OY$.
Find the area of the region bounded by the curve $f(x) = x^2 - 4x + 5$, the tangent line to this curve at the point $(3, f(3))$ and the y-axis $OY$.
Calculate the area of the region in the first quadrant enclosed by the graphs of $f(x) = \frac{8}{x}$, the line $y = x$ and the line $y = 4$.
Calculate the area of the region in the first quadrant enclosed by the graphs of $f(x) = \frac{8}{x}$, the line $y = x$ and the line $y = 4$.
A graphic design company creates a logo using the common area to the graphs of the functions $f(x) = \sin(x)$ and $g(x) = \cos(x)$ in the interval $x \in [0, \pi]$. a)…
A graphic design company creates a logo using the common area to the graphs of the functions $f(x) = \sin(x)$ and $g(x) = \cos(x)$ in the interval $x \in [0, \pi]$.
- a) Determine the points of intersection of both functions in this interval.
- b) The logo is divided into two regions by the x-axis $OX$. Calculate the area of the region of the logo that lies above the $OX$-axis.
- c) Justify whether the total area of the logo (considering absolute values) is symmetrical with respect to any value of $x$ in the given interval.
Symmetry shortcuts
Sometimes you can find an integral without finding any antiderivative.
Without calculating the antiderivative, justify using the properties of the definite integral why the value of the following integral is equal to $0$: $$\int_{-2}^{2}…
Without calculating the antiderivative, justify using the properties of the definite integral why the value of the following integral is equal to $0$:
$$\int_{-2}^{2} (x^5 + \sin(x)) \, dx$$Functions defined by an integral
An accumulation function has the form $F(x) = \displaystyle\int_a^x f(t)\,dt$. It records how much has built up from $a$ to $x$.
You never need an antiderivative of $f$ to differentiate $F$, so $F'$, tangent lines and the graph's shape come almost for free. To get $F(x_0)$, a good trick is to look for the $x_0$ where the two limits are equal, because then $F(x_0) = 0$.
Some problems ask for "the equation of the tangent line to $F$ at $x_0$". For that you need $F(x_0)$ and $F'(x_0)$.
You can also analyze $F$ like any function: $F'$ gives the intervals of increase and decrease and the relative extrema, and $F''$ gives concavity and inflection points.
Limits that involve an integral
If $F(x) = \int_0^x f(t)\,dt$ with $f$ continuous, then $F(0) = 0$ and $F'(x) = f(x)$. A limit such as $\lim_{x\to 0} \dfrac{F(x)}{x^2}$ is therefore a $\dfrac00$ form, and L'Hôpital's Rule applies. State the hypotheses in your solution. Both functions must be differentiable near the point, the form must be $\frac00$ or $\frac{\infty}{\infty}$, the derivative of the denominator must be nonzero near the point, and the limit of the quotient of derivatives must exist.
When the numerator is a product, as in $x\cdot\int_0^x g(t)\,dt$, differentiate it with the product rule. You may need L'Hôpital's Rule more than once. If you see a $\cos(x^2)$ or similar in the denominator, use the chain rule when you differentiate it.
Given the integral function $F(x) = \int_{0}^{x} \sqrt{t^2 + 9} \, dt$, use the Fundamental Theorem of Calculus to find the value of its derivative at the point $x = 4$,…
Given the integral function $F(x) = \int_{0}^{x} \sqrt{t^2 + 9} \, dt$, use the Fundamental Theorem of Calculus to find the value of its derivative at the point $x = 4$, that is, calculate $F'(4)$.
Let the function $F(x)$ be defined by the following integral: $F(x) = \int_{2}^{x^2-1} \frac{\ln(t)}{t} dt$ Find the equation of the tangent line to the graph of $F(x)$…
Let the function $F(x)$ be defined by the following integral:
$F(x) = \int_{2}^{x^2-1} \frac{\ln(t)}{t} dt$
Find the equation of the tangent line to the graph of $F(x)$ at the point with x-coordinate $x = \sqrt{3}$.
Let the function $f(x)$ be defined by the following expression involving an integral: $F(x) = \int_{0}^{x^2} \frac{t-1}{2+e^t} dt$ You are asked to determine with…
Let the function $f(x)$ be defined by the following expression involving an integral:
$F(x) = \int_{0}^{x^2} \frac{t-1}{2+e^t} dt$
You are asked to determine with reasoning:
- a) The intervals of increase and decrease of $F(x)$ in its entire domain.
- b) The x-coordinate of the points where $F(x)$ reaches its relative extrema, classifying them as maxima or minima.
- c) Does the function $F(x)$ have any points of inflection in the interval $(0, +\infty)$? Justify your answer using the second derivative.
Calculate the value of the following limit applying the Fundamental Theorem of Calculus and L'Hôpital's Rule:…
Calculate the value of the following limit applying the Fundamental Theorem of Calculus and L'Hôpital's Rule:
$\lim_{x \to 0} \frac{x \cdot \int_{0}^{x} e^{-t^2} dt}{1 - \cos(x^2)}$
State in detail the steps followed and the necessary hypotheses for the application of each theorem.
Finding an unknown parameter
Sometimes the area is given and the unknown is a constant in the function or a limit. The method is always the same:
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Set up the integral with the unknown inside. The limits are often functions of the unknown.
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Evaluate the integral with the FTC.
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Set the result equal to the given value and solve.
Determine the value of the positive parameter $k > 0$ so that the area of the region bounded by the parabola $y = kx^2$ and the line $y = 4$ is exactly $16$ square units.
Determine the value of the positive parameter $k > 0$ so that the area of the region bounded by the parabola $y = kx^2$ and the line $y = 4$ is exactly $16$ square units.
A team of engineers is designing the entrance arch to a technological tunnel. The shape of the upper arch is given by the function $f(x) = 4 - x^2$ for $x \in [-2, 2]$.…
A team of engineers is designing the entrance arch to a technological tunnel. The shape of the upper arch is given by the function $f(x) = 4 - x^2$ for $x \in [-2, 2]$. To reinforce the structure, a straight line $y = k$ (where $0 < k < 4$) must be drawn in such a way that the area of the upper section (above the reinforcement line) is exactly twice the area of the lower section (between the reinforcement line and the ground, located on the $OX$-axis).
Calculate the exact value of the constant $k$ that satisfies this condition.
Volumes of solids of revolution
When a region is rotated around a line, each thin slice becomes a disk or a washer (a disk with a hole).
Some problems call for the area first and the volume second, as in "set up, but do not evaluate" tasks. Sketch the region, find the intersection points, decide between $dx$ and $dy$, then write $R$ and $r$.
Consider the region of the plane $R$ bounded by the curve $y = \ln(x)$, the tangent line to this curve at the point with x-coordinate $x = e$, and the x-axis $OX$. a)…
Consider the region of the plane $R$ bounded by the curve $y = \ln(x)$, the tangent line to this curve at the point with x-coordinate $x = e$, and the x-axis $OX$.
- a) Find the equation of the mentioned tangent line.
- b) Calculate the area of the region $R$.
- c) Calculate the volume of the solid of revolution generated by rotating the region $R$ about the $OX$-axis.
A metal part is to be manufactured whose cross-section is bounded by the curves $y = \sqrt{x}$, $y = x - 2$ and the y-axis $OY$. a) Graphically represent the region and…
A metal part is to be manufactured whose cross-section is bounded by the curves $y = \sqrt{x}$, $y = x - 2$ and the y-axis $OY$.
- a) Graphically represent the region and determine its points of intersection.
- b) Calculate the area of this section.
- c) If the part is obtained by rotating this region about the y-axis $OY$, set up (without needing to calculate) the integral or sum of integrals that would provide the volume of the part.
Given the function $f(x) = \frac{x^2}{x^2 + 1}$, consider the region bounded by the graph of $f(x)$, its horizontal asymptote and the vertical lines $x = -1$ and…
Given the function $f(x) = \frac{x^2}{x^2 + 1}$, consider the region bounded by the graph of $f(x)$, its horizontal asymptote and the vertical lines $x = -1$ and $x = 1$.
- a) Identify the equation of the horizontal asymptote of the function.
- b) Calculate the area of this region.
- c) Determine the volume of the solid generated by rotating this region about its own horizontal asymptote.
(Hint: Perform a substitution or translation to make calculating the volume easier).
Back to the start: Aaliyah's speed log is a rate, and the distance is its integral. Since speed is never negative, the distance driven is the area under the speed graph, $\int_a^b |v(t)|\,dt$.