Until now you have been able to calculate the areas of triangles, trapeziums or circles because they have formulae. But what is the area of the region between the curve $f(x)=x^2$, the $OX$ axis and the lines $x=0$ and $x=2$? There is no geometric formula for that shape. The definite integral was created precisely to solve this problem.
The idea is simple: if you cannot calculate the exact area of a curved shape, approximate it with shapes whose area you can calculate. Divide the interval $[a,b]$ into narrow strips of width $\Delta x$ and, in each strip, draw a rectangle whose height is (approximately) the value of $f(x)$ in that strip.
The more rectangles you use (and the narrower they are), the less "extra" or "missing" area there is between the steps and the real curve, and the sum of their areas gets closer to the true area.
When $f$ is constant, the "rectangles" are all equal to the region itself: there is nothing to approximate, and the area is exact from the very first rectangle.
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In physics, the distance travelled by an object can be calculated as the area under the curve in a velocity against time graph. If an e-scooter moves at a constant…
In physics, the distance travelled by an object can be calculated as the area under the curve in a velocity against time graph. If an e-scooter moves at a constant velocity of $v(t) = 12$ km/h during the first half hour ($t = 0,5$ h), set up and calculate the definite integral that represents the distance travelled in that time.
When the number of rectangles grows without limit ($n\to\infty$ and $\Delta x\to 0$), the sum of their areas settles on a number. That number is the definite integral of $f$ between $a$ and $b$.
When $f$ is always positive on $[a,b]$, $\displaystyle\int_a^b f(x)\,dx$ is exactly the area between the curve and the $OX$ axis. If $f$ is a simple function (a straight line, for example), you can check that the integral matches the area calculated using geometry.
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Consider the linear function $f(x) = \frac{1}{2}x + 1$. You are asked 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 asked 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 formulae (area of a trapezium).
Calculate the definite integral $\int_{0}^{4} (\frac{1}{2}x + 1) \, dx$ using Barrow's rule 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, obtain 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, obtain the result of:
A car wash tunnel has a metallic structure whose upper part follows the shape of the parabola $f(x) = -x^2 + 4$. Calculate the area of the flat (front) surface bounded…
A car wash tunnel has a metallic 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 parabolic structure and the ground ($OX$-axis), knowing that the support points on the ground are $x = -2$ and $x = 2$.
Functions that change sign: the integral must be split
If $f$ is positive on one part of $[a,b]$ and negative on another, the integral $\int_a^b f(x)\,dx$ mixes positive and negative areas and does not give the total area of the shaded region (the parts can cancel out). To find the real area, first locate the points where $f$ crosses the $OX$ axis and split the integral into as many pieces as there are changes of sign.
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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: Notice 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: Notice that the function crosses the $OX$-axis at a point inside the given interval, so you will need to split the region into two sub-areas.
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, with justification, the value of:
Determine the area of the region bounded by the curve $f(x) = x^3 - 4x$ and the $OX$-axis in the interval $[-1, 2]$. Note that the function may change sign within the…
Determine the area of the region bounded by the curve $f(x) = x^3 - 4x$ and the $OX$-axis in the interval $[-1, 2]$. Note that the function may change sign within the interval, so you will need to split the integral if necessary.
Back to the start: The exact area under $f(x)=x^2$ between $x=0$ and $x=2$ is calculated with the definite integral $\int_0^2 x^2\,dx = \left[\frac{x^3}{3}\right]_0^2 = \frac{2^3}{3} - 0 = \frac{8}{3}\text{ u}^2$.
Summary
Let's practice! 10 medium exercises in this section