Evaluating Definite Integrals: \int_0^1 \sqrt{x} - x^3 dx Explained

tweety1234 · 2009-05-05T12:21:28+00:00

\int_0^1 \sqrt{x} - x^3 dx How do I evaluate this? what does \sqrt{x} -x^3 = ?

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Start date May 5, 2009
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Definite integral Integral

In summary, a definite integral is a mathematical concept used to find the total value or quantity of a function over a given interval by finding the area under a curve on a graph. It is calculated using the Fundamental Theorem of Calculus and differs from an indefinite integral in that it has specific limits of integration and gives a numerical value. This concept is important in various fields of science and engineering, as it allows for the calculation of important quantities and is used in solving differential equations and studying rates of change. Some real-world applications of definite integrals include finding displacement, determining volume, and calculating costs in fields such as physics, economics, and engineering.

May 5, 2009

tweety1234

[itex] \int_0^1 \sqrt{x} - x^3 dx [/itex]

How do I evaluate this?

what does [itex] \sqrt{x} -x^3 = [/itex] ?

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May 5, 2009

Cyosis

Homework Helper

It is [tex]\sqrt{x}+(-x^3) \Rightarrow \int_0^1 \sqrt{x}+(-x^3) dx=\int_0^1 \sqrt{x}+\int_0^1 -x^3 dx[/tex]

May 5, 2009

HallsofIvy

Science Advisor

Homework Helper

[itex]\sqrt{x}- x^3= x^{1/2}- x^3[/itex]. That should be easy to integrate using the "power rule".

What is a definite integral?

A definite integral is a mathematical concept that represents the area under a curve on a graph. It is used to find the total value or quantity of a function over a given interval.

How is a definite integral calculated?

A definite integral is calculated using the Fundamental Theorem of Calculus, which involves taking the antiderivative of the function and evaluating it at the upper and lower limits of the interval.

What is the difference between a definite and indefinite integral?

A definite integral has specific limits of integration and gives a numerical value, while an indefinite integral does not have limits and represents a general antiderivative of a function.

Why is the concept of a definite integral important?

The concept of a definite integral is important in many fields of science and engineering, as it allows us to calculate important quantities such as area, volume, and displacement. It is also used in solving differential equations and in the study of rates of change.

What are some real-world applications of definite integrals?

Definite integrals have many real-world applications, such as calculating the area under a velocity-time graph to find displacement, determining the volume of irregularly shaped objects, and finding the total cost of a changing rate of production. They are also used in physics, economics, and engineering.