The fundamental theorem of integral calculus and Barrow's rule
In the previous section you calculated some definite integrals by adding up rectangles or by comparing with geometric formulae. For an arbitrary function that is not feasible. The fundamental theorem of calculus links the integral (area) with the derivative (gradient) and finally gives you a quick method: Barrow's rule.
The function defined by an integral
Fix the lower limit $a$ and let the upper limit vary: for each $x$, $F(x) = \int_a^x f(t)\,dt$ is the area accumulated from $a$ to $x$. $F$ is a new function, with one value for each $x$.
Given a function $f(x)$ continuous on the interval $[1, 8]$, the values of the following definite integrals are known: $\int_1^5 f(x) \, dx = 7$…
Given a function $f(x)$ continuous on the interval $[1, 8]$, the values of the following definite integrals are known:
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$\int_1^5 f(x) \, dx = 7$
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$\int_5^8 f(x) \, dx = -3$
Use the properties of the definite integral to calculate the value of $\int_1^8 2 \cdot f(x) \, dx$.
The fundamental theorem of calculus
Here is the piece that connects the whole analysis unit: the derivative of this "accumulated area" function is, quite simply, the original function.
When the upper limit is not $x$ but a function of $x$ (for example $x^2$), you must also apply the chain rule.
A video game programmer is designing the trajectory of a projectile. The velocity of the object as a function of time $t$ is given by the expression…
A video game programmer is designing the trajectory of a projectile. The velocity of the object as a function of time $t$ is given by the expression $v(t) = \int_1^t (3x^2 - 4x) \, dx$.
Apply the Fundamental Theorem of Calculus to find the expression for the instantaneous acceleration $a(t)$, knowing that acceleration is the derivative of velocity with respect to time ($a(t) = v'(t)$).
Calculate the derivative function $F'(x)$ of the following integral function using the Fundamental Theorem of Calculus and the chain rule: $$F(x) = \int_{1}^{x^2}…
Calculate the derivative function $F'(x)$ of the following integral function using the Fundamental Theorem of Calculus and the chain rule:
$$F(x) = \int_{1}^{x^2} \frac{\ln(t)}{t + 1} \, dt$$State the domain of definition of the function $F(x)$ so that the integral is well-defined.
Barrow's rule
The fundamental theorem says that a definite integral can be calculated with any antiderivative: you evaluate the antiderivative at the two endpoints and subtract. You no longer need sums of rectangles.
A solar thermal panel has a design whose upper cross-section is bounded by the function $f(x) = 3x^2 + 2$, where $x$ is measured in metres. Calculate the value of the…
A solar thermal panel has a design whose upper cross-section is bounded by the function $f(x) = 3x^2 + 2$, where $x$ is measured in metres. Calculate the value of the definite integral on the interval $[0, 2]$ using Barrow's Rule.
Given the function $f(x) = |x^2 - 2x|$, calculate the value of the definite integral on the interval $[0, 3]$ using Barrow's Rule: $$\int_0^3 |x^2 - 2x| \, dx$$ Briefly…
Given the function $f(x) = |x^2 - 2x|$, calculate the value of the definite integral on the interval $[0, 3]$ using Barrow's Rule:
$$\int_0^3 |x^2 - 2x| \, dx$$Briefly explain why it is necessary to split the interval of integration to solve this exercise.
Let the function $f(t) = \frac{\ln(t)}{t^2}$ be defined for $t > 0$. A new integral function $G(x)$ is defined as: $$G(x) = \int_{x}^{x^2} f(t) \, dt$$ a) Applying the…
Let the function $f(t) = \frac{\ln(t)}{t^2}$ be defined for $t > 0$. A new integral function $G(x)$ is defined as:
$$G(x) = \int_{x}^{x^2} f(t) \, dt$$- a) Applying the Fundamental Theorem of Calculus and the chain rule, obtain the expression for the derivative $G'(x)$.
- b) Determine the intervals of increase and decrease of $G(x)$ in its domain.
- c) Find the value of $x > 1$ for which the function $G(x)$ reaches a local maximum.