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There are three valid answers. Wolfram is calculating this one:ArcanaNoir said:The real question of the day is why (-1)^(2/3) doesn't evaluate to 1.
jbunniii said:There are three valid answers. Wolfram is calculating this one:
$$(-1)^{2/3} = (e^{i\pi})^{2/3} = e^{i2\pi/3} = \cos(2\pi/3) + i\sin(2\pi/3) = -1/2 + i\sqrt{3}/2$$
These two also work:
$$(-1)^{2/3} = (e^{-i\pi})^{2/3} = e^{-i2\pi/3} = \cos(-2\pi/3) + i\sin(-2\pi/3) = -1/2 - i\sqrt{3}/2$$
$$(-1)^{2/3} = (e^{i3\pi})^{2/3} = e^{i2\pi} = 1$$
tahayassen said:Mind has now been totally blown.
The definite integral is not inherently wrong. It is a mathematical concept used to find the area under a curve or the accumulation of a function over a given interval. However, there can be errors in the calculation or interpretation of a definite integral that can make it incorrect.
If you have a specific definite integral in question, you can double-check your work and calculations to see if there were any errors. Additionally, you can use the Fundamental Theorem of Calculus to evaluate the integral and compare it to your answer. If they do not match, there may be a mistake in your work.
Some common mistakes when working with definite integrals include incorrect limits of integration, incorrect use of integration rules, miscalculations, and misinterpretation of the meaning of the integral. It is essential to follow the correct steps and pay attention to detail to avoid these errors.
Yes, a definite integral can be negative. This means that the area under the curve is below the x-axis, and it has a negative value. It is essential to pay attention to the orientation of the curve and the limits of integration to determine if the integral will be positive or negative.
Definite integrals have various real-world applications, such as calculating the total distance traveled by an object with changing velocity, finding the total amount of fluid flowing through a pipe, and determining the total amount of work done by a force over a distance. They are also used in economics, physics, and engineering to solve various problems.