Applications of the Definite Integral
A car moves along the Autopista Duarte, and the speedometer shows its speed at every instant. If you have the formula for that speed, how do you find how many meters the car traveled in one minute? With the definite integral, which you already know how to compute using Barrow's rule. In this section you apply it to areas, distances, costs, functions defined by integrals, and unknown parameters.
Area Under a Curve with Barrow's Rule
The procedure is always the same. First you find the limits of integration (if they are not given, they come from the intersection points). Then you find an antiderivative and apply Barrow's rule: $\int_a^b f(x)\,dx = G(b)-G(a)$.
If $f(x)\ge 0$ on $[a,b]$, that integral is the area between the curve and the $OX$ axis. If $f(x)\le 0$, the integral comes out negative and the area is its absolute value.
Calculate the value of the following definite integral applying Barrow's Rule: $$\int_{1}^{2} (3x^2 - 4x + 5) \, dx$$ <br>
Calculate the value of the following definite integral applying Barrow's Rule:
$$\int_{1}^{2} (3x^2 - 4x + 5) \, dx$$Without computing the antiderivative, justify using the properties of the definite integral why the value of the following integral is equal to $0$: $$\int_{-2}^{2} (x^5…
Without computing the antiderivative, justify using the properties of the definite integral why the value of the following integral is equal to $0$:
$$\int_{-2}^{2} (x^5 + \sin(x)) \, dx$$A graphic designer is drawing the outline of a logo that follows the curve $f(x) = e^x$. Calculate the area enclosed under that curve, the $OX$ axis and the vertical…
A graphic designer is drawing the outline of a logo that follows the curve $f(x) = e^x$. Calculate the area enclosed under that curve, the $OX$ axis and the vertical lines $x = 0$ and $x = \ln(3)$.
When the Curve Crosses the Axis
If the curve crosses the $OX$ axis within the interval, the integral subtracts the regions below the axis from those above it. To get the area you have to split it into pieces.
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 find the intersection points of the…
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 find the intersection points of the function with the axis first.
A design studio is designing 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 designing 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 enclosed regions formed by the curve and the axis.
Area Between Two Curves
When the region is bounded by two functions $f$ and $g$, at each $x$ the height of the region is the distance between the curves, $f(x)-g(x)$. It doesn't matter where the $OX$ axis is.
Some regions have more than one part: a vertical line like $x=1$ may cut the region, or the curves may cross in the middle. When that happens, draw the figure and compute each piece with its own "top curve" and "bottom curve."
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 enclosed by the graphs of the functions $f(x) = -x^2 + 5x$ and $g(x) = x + 2$. Sketch the region before performing the…
Calculate the area of the bounded plane region enclosed by the graphs of the functions $f(x) = -x^2 + 5x$ and $g(x) = x + 2$. Sketch the region before performing the calculation.
You can do it on paper and send a photo with your answer.
Find the area of the region bounded by the curve $f(x) = x^2 - 4x + 5$, the tangent line to that 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 that curve at the point $(3, f(3))$ and the y-axis $OY$.
Regions with Three Curves or with Crossings
Sometimes the region is bounded by three curves, or the two functions cross inside the interval. It is solved like this:
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Draw the curves and mark all the intersection points.
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Divide the region into vertical pieces where the top curve and the bottom curve do not change.
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On each piece integrate (top − bottom) and add the areas.
For example, the curves $y=\sin x$ and $y=\cos x$ on $[0,\pi]$ intersect at $x=\tfrac{\pi}{4}$. To the left, cosine is on top, and to the right, sine is. The integral has to be split at that point. If you are also asked to separate the part above the $OX$ axis, you must split at the intersections with the axis as well.
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$.
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 that…
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 that region.
You can do it on paper and send a photo with your answer.
A graphic design company creates a logo shaped like the region bounded by the graphs of the functions $f(x) = \sin(x)$ and $g(x) = \cos(x)$ and by the vertical lines…
A graphic design company creates a logo shaped like the region bounded by the graphs of the functions $f(x) = \sin(x)$ and $g(x) = \cos(x)$ and by the vertical lines $x = 0$ and $x = \pi$ (that is, on the interval $x \in [0, \pi]$). The figure shows both graphs and the shaded logo.
- a) Determine the points where the two functions intersect in that interval.
- b) The logo is divided into two regions by the x-axis $OX$. Calculate the area of the part of the logo that lies above the $OX$ axis.
- c) Justify whether the whole logo (the region between the two graphs on $[0, \pi]$) is symmetric with respect to some vertical line $x = a$ with $a \in [0, \pi]$.
The Integral as an Accumulated Quantity
If a quantity changes at a certain rate, the total accumulated between two instants is the integral of that rate.
With velocity you have to distinguish two ideas. The integral $\int_a^b v(t)\,dt$ is the displacement (final position minus initial position, with sign). The integral $\int_a^b |v(t)|\,dt$ is the total distance traveled. If $v$ changes sign, they are not the same.
With marginal costs it works the same way. If $C'(x)$ is the cost of producing one more unit, the increase in cost when going from $a$ to $b$ units is $\int_a^b C'(x)\,dx$.
Back to the beginning: the distance traveled by the car on the Autopista Duarte is the area under the graph of its velocity, computed with the definite integral of $v(t)$.
The velocity of a radio-controlled car (in $m/s$) is given by the function $v(t) = 2t + 1$. Determine the total distance travelled by the car between the times $t = 1$ s…
The velocity of a radio-controlled car (in $m/s$) is given by the function $v(t) = 2t + 1$. Determine the total distance travelled by the car between the times $t = 1$ s and $t = 4$ s by calculating the definite integral of the velocity.
An electronics company has determined that the marginal cost of producing $x$ units of a sensor is $C'(x) = 180x^2 - 720x + 900$ pesos per unit. Calculate the total…
An electronics company has determined that the marginal cost of producing $x$ units of a sensor is $C'(x) = 180x^2 - 720x + 900$ pesos per unit. Calculate the total increase in cost when increasing production from $10$ to $20$ sensors.
An autonomous vehicle travels in a straight line and its speed (in m/s) is given by the function $v(t) = t^2 - 6t + 8$, where $t$ is the time in seconds. Calculate the…
An autonomous vehicle travels in a straight line and its speed (in m/s) is given by the function $v(t) = t^2 - 6t + 8$, where $t$ is the time in seconds. Calculate the total distance travelled by the vehicle between time $t = 0$ and $t = 5$. Note: Remember that the distance is the integral of the modulus of the velocity.
Functions Defined by an Integral: Derivative and Tangent
If $F(x)=\int_a^x f(t)\,dt$, the Fundamental Theorem of Calculus says that $F'(x)=f(x)$. You don't need the antiderivative to differentiate $F$.
When the upper limit is not $x$ but a function $g(x)$, the chain rule is used:
$$F(x)=\int_a^{g(x)} f(t)\,dt \;\Longrightarrow\; F'(x)=f\big(g(x)\big)\cdot g'(x)$$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 $x = \sqrt{3}$.
Studying a Function Defined by an Integral
Since $F'(x)=f(g(x))\,g'(x)$ is known, you can study $F$ just like any other function. The sign of $F'$ gives where it increases and decreases. The points where $F'$ goes from $+$ to $-$ are maxima, and from $-$ to $+$ are minima.
For limits with an integral inside (of the $0/0$ type), L'Hôpital's rule lets you differentiate the numerator and denominator separately. The Fundamental Theorem gives you the derivative of the integral.
Let the function $f(x)$ be defined by the following expression involving an integral: $F(x) = \int_{0}^{x^2} \frac{t-1}{2+e^t} dt$ Determine, giving reasons: a) The…
Let the function $f(x)$ be defined by the following expression involving an integral:
$F(x) = \int_{0}^{x^2} \frac{t-1}{2+e^t} dt$
Determine, giving reasons:
- a) The intervals of increase and decrease of $F(x)$ across its entire domain.
- b) The x-coordinates of the points where $F(x)$ reaches its local extrema, classifying them as maximums or minimums.
- c) Does the function $F(x)$ have any point 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:
$\lim_{x \to 0} \frac{x \times \int_{0}^{x} e^{-t^2} dt}{1 - \cos(x^2)}$
State in detail the steps followed and the necessary hypotheses for applying each theorem.
Finding a Parameter or a Time from an Area
Many problems give the value of an area (or a distance) and ask for a constant or an instant. The method is always: set up the integral with the unknown inside, apply Barrow's rule, set it equal to the given value, and solve.
When the condition is a ratio between two areas (for example, "one is twice the other"), compute the total area and express the part that depends on the parameter. Then set up the equation between the two parts. For $f(x)=4-x^2$ and a line $y=k$, the upper part is $\int_{-\sqrt{4-k}}^{\sqrt{4-k}}(4-x^2-k)\,dx=\tfrac43(4-k)^{3/2}$.
For an instant $T$ that splits the distance in two, use the accumulated distance $D(T)=\int_0^T|v(t)|\,dt$ and set it equal to half of the total. Pay close attention to which piece $T$ falls in, because $|v|$ changes expression at the points where $v=0$.
Determine the value of the positive parameter $k > 0$ such that the area of the region bounded by the parabola $y = kx^2$ and the line $y = 4$ is exactly $16$ square…
Determine the value of the positive parameter $k > 0$ such 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 archway to a technology 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 archway to a technology 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 such that the area of the upper section (above the reinforcing line) is exactly twice the area of the lower section (between the reinforcing line and the ground, located on the $OX$ axis).
Calculate the exact value of the constant $k$ that satisfies this condition.
A surveillance drone moves in a straight line. Its velocity in meters per second is given by the absolute value function: $v(t) = |t^2 - 6t + 5|$ for $t \in [0, 6]$…
A surveillance drone moves in a straight line. Its velocity in meters 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 the total distance calculated in the previous part.
Areas Integrating with Respect to y
Some regions are better described with horizontal slices. Then you express each curve as $x=h(y)$, and the height of the region becomes its width: $x_{\text{right}}-x_{\text{left}}$.
A metal piece is to be manufactured whose cross section is bounded by the curves $y = \sqrt{x}$, $y = x - 2$ and the $y$-axis $OY$. a) Sketch the region and determine…
A metal piece is to be manufactured whose cross section is bounded by the curves $y = \sqrt{x}$, $y = x - 2$ and the $y$-axis $OY$.
- a) Sketch the region and determine its points of intersection.
- b) Calculate the area of that cross section.
- c) If the piece 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 give the volume of the piece.
You can do it on paper and send a photo with your answer.
The Horizontal Asymptote as a Limit
Some regions are bounded by a curve and its horizontal asymptote: a line $y=L$ that the graph gets closer and closer to when $x$ becomes very large, positive or negative.
To compute it, divide the numerator and denominator by the highest power of $x$. Terms like $\dfrac1{x^2}$ tend to $0$ as $x\to\pm\infty$.
A Look at Volumes of Revolution
If you revolve a plane region around an axis, you get a solid. When you revolve the region under $y=f(x)$ on $[a,b]$ around the $OX$ axis, each vertical slice is a disk of radius $f(x)$.
If the axis of revolution is a horizontal line $y=c$, the radius is the distance from the curve to that line, $|f(x)-c|$. If the axis is $OY$, you integrate with respect to $y$ with the curves as $x=h(y)$ and use $\pi\int [h(y)]^2\,dy$ (with washers if there is a hole). If the region has two different parts, as in the section on integrating with respect to $y$, you add two integrals.
In the exercises below, besides the volume, you will find the equation of the tangent and the area. Always start by drawing the region.
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 aforementioned 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.
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 around its horizontal asymptote.
(Hint: Perform a change of variable or translation to facilitate the calculation of the volume).