What if we integrate wrt to y instead of x

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Integrating with respect to y instead of x involves treating x as a constant if it does not depend on y. The integral of a constant x with respect to y results in xy, similar to integrating a constant c. If x is a function of y, it should be expressed in terms of y before integration. This approach clarifies the process of integration when the variable of integration differs from the function being integrated. Understanding these principles enhances mathematical problem-solving skills.
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Hi guys, i was doing some math qn and i found out that i am always integrating like x wrt x so u will get (x^2)/2 right? So i was wondering how do u like say integrate x wrt y?
 
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If x is a constant with respect to y, then the integral of x with respect to y is just xy (just like the integral of c with respect to y is cy when c is a constant). Otherwise, if x depends on y (ie, x=x(y)), then write out x in terms of y and do the integral as usual.
 
alright got it thx
 
Question: A clock's minute hand has length 4 and its hour hand has length 3. What is the distance between the tips at the moment when it is increasing most rapidly?(Putnam Exam Question) Answer: Making assumption that both the hands moves at constant angular velocities, the answer is ## \sqrt{7} .## But don't you think this assumption is somewhat doubtful and wrong?

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