Applications of the definite integral
You now have the tools: you know what a definite integral is, its properties, and Barrow's rule for calculating it. Now it's time to use them to solve the problems that really come up in the exam: areas between curves, quantities that accumulate over time, and unknown parameters that can be deduced from an area.
Area under a curve using Barrow's rule
The procedure is always the same: find the limits of integration (if you are not given them, find them using the points of intersection), find an antiderivative and apply Barrow's rule. If the region lies below the axis, remember to take the 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:
$$\int_{1}^{2} (3x^2 - 4x + 5) \, dx$$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.
Area between two curves
When the region is bounded by two functions $f$ and $g$ instead of by the $OX$ axis, the idea is the same but with a subtraction: at each $x$, the "height" of the region is the distance between the two curves.
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 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 $g(x) = 2$.
The integral as an accumulated quantity
If a quantity changes at a certain "speed" or "rate", the total accumulated between two instants is the integral of that rate. It is the same idea you have already used with antiderivatives, now with specific limits.
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 $C'(x) = 3x^2 - 12x + 15$ euros per unit. Calculate the total increase…
An electronics company has determined that the marginal cost of producing $x$ units of a sensor is $C'(x) = 3x^2 - 12x + 15$ euros per unit. Calculate the total increase in cost when increasing production from $10$ to $20$ sensors.
Functions defined by an integral: tangent lines
Remember from the previous section that if $F(x)=\int_a^x f(t)\,dt$, then $F'(x)=f(x)$ by the fundamental theorem. This allows you to find the tangent line to $F$ without calculating any antiderivative: you only need $F$ at one point (an integral with numerical limits) and $F'$ at that point (the value of $f$ there).
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 $x = \sqrt{3}$.
Finding a parameter from an area
Many problems give you the value of the area and ask for a constant in the function or in the limits of integration. The method: set up the integral with the unknown inside, apply Barrow's rule, and set the result equal to the given area.
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.