Yes, but you're saying it in kind of a confusing way. When you say "x going from 2x to 1," you're using "x" to mean two different things. It would be better to say x goes from y/2 to 1.Kuma said:
Yes, that's right. The triangle is bounded on two sides by the x-axis and the line x=1. The third side is the line y=2x or, equivalently, x=y/2. The first form is useful when you integrate with respect to y first; the second form is useful when you integrate with respect to x first.
Changing bounds of integration refers to the process of modifying the limits of integration in a definite integral. This is often done to simplify the calculation or to change the region of integration.
There are several reasons why someone might need to change the bounds of integration. One common reason is to simplify the calculation by choosing more convenient limits. Another reason is to change the region of integration, such as when dealing with multivariable functions.
To change the bounds of integration, you first need to identify the new limits. This can be done by manipulating the original limits or by graphing the function and determining the appropriate limits. Once you have the new limits, you can substitute them into the integral and proceed with the calculation.
Yes, changing the bounds of integration can affect the value of the integral. This is because the limits of integration determine the region over which the function is being integrated. By changing the limits, you may be including or excluding certain parts of the function, leading to a different result.
Yes, there are some limitations to changing the bounds of integration. For example, the new limits must still be within the original limits of integration. Additionally, the new limits must be chosen in a way that does not change the nature of the integral, such as swapping the upper and lower limits. It is important to carefully consider the implications of changing the bounds before proceeding with the calculation.