Area under a curve
What is the area of the region between the curve $f(x)=e^x$, the $OX$ axis and the lines $x=0$ and $x=1$? There is no geometric formula for that curved shape. The definite integral was invented precisely to solve this kind of problem: approximate the area with rectangles and, in the limit, calculate it using an antiderivative.
Approximating the area with rectangles
The idea behind the definite integral is to divide the region into very narrow strips and add up their areas. When the function is constant, you do not even need the limit: the area is exact from the very first "rectangle".
Find the area of the region bounded by the constant function $f(x) = 4$, the $OX$-axis and the lines $x = -1$ and $x = 2$. Justify the result graphically by calculating…
Find the area of the region bounded by the constant function $f(x) = 4$, the $OX$-axis and the lines $x = -1$ and $x = 2$. Justify the result graphically by calculating the area of the resulting rectangle without using integrals.
Area under the curve using Barrow's rule
The usual procedure: if $f(x)\geq0$ throughout $[a,b]$, the area is simply the definite integral, calculated using Barrow's rule.
An athlete is training for a marathon. Their speed over a 15-minute stretch is constant and given by the function $v(t) = 12$ km/h, where $t$ is time in hours. Use the…
An athlete is training for a marathon. Their speed over a 15-minute stretch is constant and given by the function $v(t) = 12$ km/h, where $t$ is time in hours. Use the definite integral to calculate the distance travelled (the area under the speed curve) between the times $t = 0$ and $t = 0,25$ hours. Check the result using the geometric formula for the area of a rectangle.
Calculate the area of the plane region bounded by the parabola $f(x) = x^2 + 2$, the x-axis and the lines $x = 0$ and $x = 2$. Since the function is always positive in…
Calculate the area of the plane region bounded by the parabola $f(x) = x^2 + 2$, the x-axis and the lines $x = 0$ and $x = 2$. Since the function is always positive in that interval, set up the definite integral directly using Barrow's rule.
Given the polynomial function $f(x) = 3x^2 + 2x$, find the area bounded by its graph and the $OX$-axis in the interval $[1, 2]$. Check that, in this interval, the…
Given the polynomial function $f(x) = 3x^2 + 2x$, find the area bounded by its graph and the $OX$-axis in the interval $[1, 2]$. Check that, in this interval, the function does not cross the $OX$-axis (that is, it is always positive), which facilitates direct calculation.
The sign of the integral: splitting into sections
If $f$ changes sign inside $[a,b]$, the integral over the whole interval mixes positive and negative parts and does not give the real area: you have to locate where $f$ equals zero and split the integral.
Find the area of the region bounded by the curve $f(x) = x^3 - 4x$ and the x-axis. Take into account that the function can take positive and negative values, so you will…
Find the area of the region bounded by the curve $f(x) = x^3 - 4x$ and the x-axis. Take into account that the function can take positive and negative values, so you will need to calculate the area of each region separately and sum their absolute values.
Given the piecewise-defined function: $f(x) = \begin{cases} 2x & \text{if } 0 \le x \le 1 \\ 3 - x & \text{if } 1 < x \le 3 \end{cases}$ Calculate the area of the region…
Given the piecewise-defined function:
$f(x) = \begin{cases} 2x & \text{if } 0 \le x \le 1 \\ 3 - x & \text{if } 1 < x \le 3 \end{cases}$
Calculate the area of the region bounded by the graph of $f(x)$ and the x-axis in the interval $[0, 3]$.
Find the area of the region bounded by the function $f(x) = |x^2 - 4|$, the x-axis and the lines $x = -1$ and $x = 3$. Tip: Identify the intervals where the expression…
Find the area of the region bounded by the function $f(x) = |x^2 - 4|$, the x-axis and the lines $x = -1$ and $x = 3$.
Tip: Identify the intervals where the expression inside the absolute value changes sign and split the definite integral into the sum of several integrals accordingly.
Applications: revenue, parameters and accumulated quantities
The area under a curve and the "total accumulated amount" are, at heart, the same calculation when the function represents a rate (marginal revenue, consumption, saving).
The marketing department of a new music app has determined that the marginal revenue per thousand new subscribers is $I'(x) = 10 - x$ (in hundreds of euros), where $x$…
The marketing department of a new music app has determined that the marginal revenue per thousand new subscribers is $I'(x) = 10 - x$ (in hundreds of euros), where $x$ is the number of thousands of subscribers. Calculate the total accumulated profit (area under the curve) when the number of subscribers increases from $x = 1$ thousand to $x = 4$ thousand.
Find the value of a positive parameter $k$ such that the area of the region bounded by the function $f(x) = kx^2$, the $OX$-axis and the vertical line $x=3$ is equal to…
Find the value of a positive parameter $k$ such that the area of the region bounded by the function $f(x) = kx^2$, the $OX$-axis and the vertical line $x=3$ is equal to 18 square units.
The marginal revenue function of an agricultural cooperative is given by $I'(x) = 100 - 0,2x^2$ (in euros per unit sold), where $x$ represents the number of tonnes of…
The marginal revenue function of an agricultural cooperative is given by $I'(x) = 100 - 0,2x^2$ (in euros per unit sold), where $x$ represents the number of tonnes of product.
- a) Calculate the total revenue function $I(x)$ knowing that for zero sales, revenue is zero.
- b) Determine the area enclosed by the curve $I'(x)$ and the x-axis in the interval where the marginal revenue is positive. Explain the economic meaning of this value.
Back to the beginning: The exact area under $f(x)=e^x$ between $0$ and $1$ is $\displaystyle\int_0^1 e^x\,dx = \big[e^x\big]_0^1 = e^1 - e^0 = e - 1 \approx 1{,}72\text{ u}^2$.