Emma is designing a curved wall ramp for a skate park. Its side profile is the curve $y = x^2$ (with $x$ and $y$ in feet) from $x = 0$ to $x = 2$, and she needs the area of that profile to order paint. There is no geometry formula for a shape with a curved top. The definite integral was invented to solve exactly this problem.
Rectangles and the Definite Integral
If you can't find the exact area of a curved shape, approximate it with shapes you can measure. Split $[a, b]$ into $n$ narrow strips of equal width $\Delta x = \dfrac{b-a}{n}$. In each strip, build a rectangle whose height is the value of $f$ at one point of that strip.
The more rectangles you use, the narrower they get and the less gap there is between the steps and the curve. The total area of the rectangles gets closer to the true area.
Now let the number of rectangles grow without bound. The sums settle down to a single number, and that number is the definite integral.
The integral also measures accumulation. A rate times a length of time is the area of a rectangle on a rate graph. If Marcus rides at a constant $15$ mph for $2$ hours, his velocity graph is a rectangle $2$ hours wide and $15$ mph tall, so the distance is $\int_0^2 15\,dt = 15 \cdot 2 = 30$ miles. The units of the answer are (units of $f$) $\times$ (units of $x$): mph $\times$ hours = miles.
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In physics, the distance traveled by an object can be calculated as the area under the curve on a velocity-time graph. If an electric skateboarder moves at a constant…
In physics, the distance traveled by an object can be calculated as the area under the curve on a velocity-time graph. If an electric skateboarder moves at a constant speed of $v(t) = 12$ mph during the first half hour ($t = 0,5$ h), set up and calculate the definite integral that represents the distance traveled in that time.
Here is the shortcut that makes integrals practical. Suppose $F'(x) = f(x)$. Then $F(b) - F(a)$ is the net change of $F$ from $a$ to $b$, and $f$ is the rate at which $F$ changes. Adding up (rate) $\times$ (tiny width) gives the total change, which is exactly what the integral does.
Here are the antiderivatives you need on this page.
$f(x)$
An antiderivative $F(x)$
$x^n$ with $n \ne -1$ (including $\sqrt{x} = x^{1/2}$ and $\frac{1}{x^2} = x^{-2}$)
$\dfrac{x^{n+1}}{n+1}$
$\dfrac{1}{x}$
$\ln\lvert x\rvert$
$e^x$
$e^x$
$\sin x$
$-\cos x$
$\cos x$
$\sin x$
$\ln x$
$x\ln x - x$
You can always check an antiderivative by differentiating it. For example, $\dfrac{d}{dx}(x\ln x - x) = \ln x + 1 - 1 = \ln x$.
Constants come out front, and sums are integrated term by term.
Check it against geometry. The region under $f(x) = 2x$ from $0$ to $3$ is a triangle with base $3$ and height $6$, so its area is $\tfrac{1}{2}(3)(6) = 9$. The integral gives $\big[x^2\big]_0^3 = 9$. Both methods agree.
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Consider the linear function $f(x) = \frac{1}{2}x + 1$. You are required to calculate the area of the shaded region in the graph, bounded by the function, the $OX$-axis…
Consider the linear function $f(x) = \frac{1}{2}x + 1$. You are required to calculate the area of the shaded region in the graph, bounded by the function, the $OX$-axis and the vertical lines $x = 0$ and $x = 4$. To do this:
Calculate the value of the area using elementary geometric formulas (area of a trapezoid).
Calculate the definite integral $\int_{0}^{4} (\frac{1}{2}x + 1) \, dx$ by applying the Fundamental Theorem of Calculus and check that both results coincide.
Calculate the value of the definite integral of the polynomial function $f(x) = 3x^2 - 2x + 5$ in the interval $[1, 2]$. That is, find the result of: $$\int_{1}^{2}…
Calculate the value of the definite integral of the polynomial function $f(x) = 3x^2 - 2x + 5$ in the interval $[1, 2]$. That is, find the result of:
The integral is built from sums, so it inherits the rules of sums. These rules let you break a hard integral into easier pieces.
Picture the area under this curve from $a$ to $b$. If you pick any point $c$ in between, the region splits into two pieces and the two areas add up to the whole. That is the "adding intervals" rule.
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Use the properties of the definite integral to solve the following problem: Given that $\int_{1}^{4} f(x) \, dx = 7$ and that $\int_{4}^{6} f(x) \, dx = -3$, calculate,…
Use the properties of the definite integral to solve the following problem:
Given that $\int_{1}^{4} f(x) \, dx = 7$ and that $\int_{4}^{6} f(x) \, dx = -3$, calculate, showing your reasoning, the value of:
A Riemann sum can't tell "above" from "below." If $f(x) < 0$, each rectangle has a negative height, so it adds a negative number to the sum.
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A graphic design company wants to create a logo that includes a curved shape defined by $f(x) = \sqrt{x}$. Calculate the area of the part of the logo enclosed between…
A graphic design company wants to create a logo that includes a curved shape defined by $f(x) = \sqrt{x}$. Calculate the area of the part of the logo enclosed between this curve, the $OX$-axis and the vertical line $x = 9$.
A car wash has a metal structure whose upper part follows the shape of the parabola $f(x) = -x^2 + 4$. Calculate the area of the flat (front) surface bounded by the…
A car wash has a metal structure whose upper part follows the shape of the parabola $f(x) = -x^2 + 4$. Calculate the area of the flat (front) surface bounded by the structure of the parabola and the ground ($OX$-axis), given that the support points on the ground are $x = -2$ and $x = 2$.
A graphic design company wants to create a logo that includes the shape of a "wave". This shape is defined by the function $f(x) = \sin(x) + 1$ in the interval…
A graphic design company wants to create a logo that includes the shape of a "wave". This shape is defined by the function $f(x) = \sin(x) + 1$ in the interval $[0, \pi]$. Calculate the total area of the figure bounded by the curve, the $OX$-axis and the vertical lines $x = 0$ and $x = \pi$.
Often a problem says only "the region bounded by the graph of $f$ and the $x$-axis," with no vertical lines. Then the curve and the axis enclose the region by themselves, and the limits of integration are the $x$-intercepts.
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Find the area of the plane region bounded by the graph of the function $f(x) = x^2 - 9$ and the x-axis ($OX$). Hint: First you must find the points of intersection of…
Find the area of the plane region bounded by the graph of the function $f(x) = x^2 - 9$ and the x-axis ($OX$).
Hint: First you must find the points of intersection of the function with the $OX$-axis to determine the limits of integration. Remember that, if the function is negative in the interval, the area is the absolute value of the integral.
Calculate the area of the plane region bounded by the graph of the function $f(x) = -x^2 + 6x - 5$ and the x-axis $OX$. Graph the region before performing the…
Calculate the area of the plane region bounded by the graph of the function $f(x) = -x^2 + 6x - 5$ and the x-axis $OX$. Graph the region before performing the calculations to identify the points of intersection.
An architect is designing the entrance to a tunnel whose front section can be modeled by the function $f(x) = 4 - \frac{1}{4}x^2$, where $x$ and $y$ are expressed in…
An architect is designing the entrance to a tunnel whose front section can be modeled by the function $f(x) = 4 - \frac{1}{4}x^2$, where $x$ and $y$ are expressed in feet. The floor of the tunnel coincides with the $OX$-axis. Calculate the area of the cross-section of the tunnel entrance.
Suppose $f$ is positive on part of $[a, b]$ and negative on another part. Then $\int_a^b f(x)\,dx$ mixes positive and negative contributions, and they can cancel. That number is the net (signed) area. It is not the total area of the shaded region. To get the total area, find where $f$ crosses the axis and split the integral there.
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Calculate the total area of the region bounded by the line $f(x) = x - 2$, the $OX$-axis and the vertical lines $x = 0$ and $x = 3$. Note: Observe that the function…
Calculate the total area of the region bounded by the line $f(x) = x - 2$, the $OX$-axis and the vertical lines $x = 0$ and $x = 3$.
Note: Observe that the function intersects the $OX$-axis at a point within the given interval, so you will have to split the region into two sub-areas.
Determine the area of the region bounded by the curve $f(x) = x^3 - 4x$ and the $OX$-axis in the interval $[-1, 2]$. Bear in mind that the function may change sign…
Determine the area of the region bounded by the curve $f(x) = x^3 - 4x$ and the $OX$-axis in the interval $[-1, 2]$. Bear in mind that the function may change sign within the interval, so you will need to split the integral if necessary.
The $x$-axis is just the function $y = 0$. Replace it with any other curve and the same idea works. Each thin vertical strip has height (top curve) $-$ (bottom curve) and width $\Delta x$.
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Calculate the area of the region bounded by the curve $f(x) = \sqrt{x}$, the $OX$-axis and the line $y = -x + 6$ in the first quadrant. Hint: first identify the point of…
Calculate the area of the region bounded by the curve $f(x) = \sqrt{x}$, the $OX$-axis and the line $y = -x + 6$ in the first quadrant. Hint: first identify the point of intersection between the curve and the line to determine the limits of integration.
Find the area of the region bounded by the function $f(x) = x^3 - 3x^2 + 2x$ and the horizontal line passing through the relative maximum of this function. Hint: You…
Find the area of the region bounded by the function $f(x) = x^3 - 3x^2 + 2x$ and the horizontal line passing through the relative maximum of this function.
Hint: You must first locate the relative maximum using the derivative, find the value of the y-coordinate at this point and then determine the points of intersection between the curve and the resulting line.
Consider the functions $f(x) = e^x$ and $g(x) = e^{-x}$. Let $R$ be the region bounded by the graphs of both functions and the vertical line $x = k$ (where $k > 0$).…
Consider the functions $f(x) = e^x$ and $g(x) = e^{-x}$. Let $R$ be the region bounded by the graphs of both functions and the vertical line $x = k$ (where $k > 0$).
Determine the value of the positive parameter $k$ so that the area of this region is exactly $4$ square units.
To integrate $|g(x)|$, rewrite it without bars. It equals $g(x)$ where $g \ge 0$ and $-g(x)$ where $g < 0$. Find the zeros of $g$, split the interval there, and integrate each formula on its own interval. A piecewise-defined function works the same way: use each formula only on the interval where it applies.
For a piecewise function such as $f(x) = x$ on $[0, 2)$ and $f(x) = 4 - x$ on $[2, 4]$, integrate each formula on its own piece: $\int_0^2 x\,dx + \int_2^4 (4 - x)\,dx = 2 + 2 = 4$. This matches the triangle with base $4$ and height $2$.
Absolute values also show up when a region is bounded by a line and an absolute-value curve. To find where they meet, solve $|g(x)| = k$ as two equations, $g(x) = k$ and $g(x) = -k$, and keep only solutions that actually make sense.
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Find the area of the region bounded by the function $f(x) = |x^2 - 1|$ and the x-axis between $x = 0$ and $x = 2$. Remember that the definition of the absolute value…
Find the area of the region bounded by the function $f(x) = |x^2 - 1|$ and the x-axis between $x = 0$ and $x = 2$. Remember that the definition of the absolute value implies that the function is always non-negative, but its analytical expression changes at the points where the argument is zero.
The function $f(x) = |x^2 - 4|$ is defined. We want to study the region of the plane that is "trapped" between this curve and the horizontal line $y = 5$. Calculate the…
The function $f(x) = |x^2 - 4|$ is defined. We want to study the region of the plane that is "trapped" between this curve and the horizontal line $y = 5$.
Calculate the area of the region bounded above by the line $y = 5$ and below by the graph of the function $f(x)$. Note: Take into account the points of intersection and the definition of the absolute value function to split the integral if necessary.
Many problems use a tangent or normal line as one of the boundaries. Recall how to get these lines at $x = a$.
Once you have the line, treat it as another curve. Sketch the region, decide which function is on top on each piece, and integrate top minus bottom. The line and the curve touch at the point of tangency, which often gives you one of the limits. The figure in an exercise tells you which boundaries enclose the region. Use it if the statement doesn't name a left or right boundary.
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Consider the function $f(x) = x^2 - 2x + 2$. Calculate the area of the region bounded by the graph of $f(x)$, the $OX$-axis and the tangent line to the curve at the…
Consider the function $f(x) = x^2 - 2x + 2$. Calculate the area of the region bounded by the graph of $f(x)$, the $OX$-axis and the tangent line to the curve at the point with x-coordinate $x = 3$.
An architect is designing the side profile of a new sports hall. The main structure follows the curve of the function $f(x) = -x^2 + 6x$. To ensure stability, a metal…
An architect is designing the side profile of a new sports hall. The main structure follows the curve of the function $f(x) = -x^2 + 6x$. To ensure stability, a metal reinforcement must be placed following the tangent line to the curve at the point with x-coordinate $x = 4$.
Calculate the area of the plane region bounded by the graph of the function $f(x)$, the aforementioned tangent line and the y-axis ($Eje\ OY$).
Consider the function $f(x) = \ln(x)$. Find: 1. Find the equation of the normal line to the graph of $f(x)$ at the point $P(1, 0)$. 2. Calculate the area of the plane…
Consider the function $f(x) = \ln(x)$. Find:
Find the equation of the normal line to the graph of $f(x)$ at the point $P(1, 0)$.
Calculate the area of the plane region bounded by the curve $y = \ln(x)$, the normal line found in the previous section and the x-axis ($Eje\ OX$).
Sometimes the area is given and you must find an unknown: a constant $k$ in the function, an endpoint $a$ or $k$, or a vertical line $x = c$ that cuts a region into pieces. The method is always the same. Write the area as an expression in the unknown, set it equal to the given value, and solve. Then check the answer against the restrictions in the problem, such as $k > 0$ or $a > 1$.
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Find the value of a positive parameter $k$ given that the area of the region bounded by the parabola $f(x) = kx^2$ and the $OX$-axis between $x = 0$ and $x = 3$ is…
Find the value of a positive parameter $k$ given that the area of the region bounded by the parabola $f(x) = kx^2$ and the $OX$-axis between $x = 0$ and $x = 3$ is exactly $18$ square units.
Let the region $A$ be the region in the first quadrant bounded by the parabola $y = x^2$ and the line $y = 9$. There is a vertical line with equation $x = c$ (with…
Let the region $A$ be the region in the first quadrant bounded by the parabola $y = x^2$ and the line $y = 9$. There is a vertical line with equation $x = c$ (with $0 < c < 3$) that divides the region $A$ into two subregions of identical areas.
Back to the start: Emma's ramp profile is the region under $y = x^2$ from $x = 0$ to $x = 2$. Its exact area is $\displaystyle\int_0^2 x^2\,dx = \left[\frac{x^3}{3}\right]_0^2 = \frac{8}{3} \approx 2.67$ ft². This sits between our left sum ($1.75$) and right sum ($3.75$) from the first section.
Summary
Let's practice! 8 medium exercises in this section