- #1
DaalChawal
- 87
- 0
How Large Can This Integral Be?
- MHB
- Thread starter DaalChawal
- Start date
In summary, a definite integral is a mathematical concept used to find the area under a curve on a graph. It is calculated by taking the antiderivative of a function and plugging in the upper and lower limits. It differs from an indefinite integral in that it gives a numerical value and has specific limits. Definite integrals have various applications in math and science and their results can be interpreted as the exact value or net change of a function over a given interval.
- #2
Opalg
Gold Member
MHB
- 2,778
- 13
$|\cos(2017\theta)|\leqslant1$.
$|5-4\cos\theta|\geqslant1$.
Use those facts to estimate how large $\displaystyle\left|\int_0^\pi\frac{\cos(2017\theta)}{5-4\cos\theta}d\theta\right|$ can be. Then decide which of the four given choices seems most likely to be correct.
$|5-4\cos\theta|\geqslant1$.
Use those facts to estimate how large $\displaystyle\left|\int_0^\pi\frac{\cos(2017\theta)}{5-4\cos\theta}d\theta\right|$ can be. Then decide which of the four given choices seems most likely to be correct.
FAQ: How Large Can This Integral Be?
What is a definite integral?
A definite integral is a mathematical concept used to calculate the area under a curve or the signed area between a curve and the x-axis on a given interval. It is represented by the symbol ∫ and has a lower and upper limit that define the interval of integration.
How is a definite integral different from an indefinite integral?
A definite integral has limits of integration and gives a specific numerical value as the result, while an indefinite integral does not have limits and gives a function as the result. In other words, a definite integral calculates a specific area, while an indefinite integral finds the general antiderivative of a function.
What is the fundamental theorem of calculus and how is it related to definite integrals?
The fundamental theorem of calculus states that the definite integral of a function f(x) can be calculated by finding the antiderivative of f(x) and evaluating it at the upper and lower limits of integration. In other words, it provides a shortcut for calculating definite integrals without having to use the limit definition.
How do you evaluate a definite integral?
To evaluate a definite integral, you can use the fundamental theorem of calculus or the limit definition. The limit definition involves dividing the interval of integration into smaller and smaller subintervals and taking the limit as the subintervals approach zero. The fundamental theorem of calculus involves finding the antiderivative of the function and evaluating it at the upper and lower limits of integration.
What are some real-life applications of definite integrals?
Definite integrals have many practical applications, such as calculating the area under a velocity-time graph to find the displacement of an object, finding the work done by a variable force, and calculating the volume of irregularly shaped objects. They are also used in economics to calculate consumer surplus and in physics to calculate electric and magnetic fields.