What does it mean by respect to Y?
The derivative with respect to x is: “at what rate does f change as x changes”, in this case it is a constant, 1. At what rate does f change as y changes, i.e. “the derivative with respect to y”, which goes like 2y.
What does in respect to Y mean?
0. Differentiating x with respect to y gives you the gradient of the tangent at the point (x,y(x)) on the curve y(x). The gradient of the tangent at the point (x,y(x)) indicates the rate of change of y(x) at that specific point.
What’s an example of horizontal?
The definition of horizontal is something that is parallel to the horizon (the area where the sky seems to meet the earth). An example of a horizontal line is one that goes across the paper. Something, such as a line, plane, or object, that is horizontal. Position parallel to the horizon.
Is area between curves always positive?
Finally, unlike the area under a curve that we looked at in the previous chapter the area between two curves will always be positive.
Can you integrate with respect to multiple variables?
The multiple integral is a type of definite integral extended to functions of more than one real variable—for example, f(x,y) f ( x , y ) or f(x,y,z) f ( x , y , z ) .
Why does the inner integral need to have y y limits?
Since the d y d y is the inner differential ( i.e. we are integrating with respect to y y first) the inner integral needs to have y y limits. Remember that we treat the x x as a constant when doing the first integral and we don’t do any integration with it yet.
Is the constant of integration a function of Y Y?
Likewise, in the second integral, the “constant” of integration must be a function of y y since we are integrating with respect to x x. Again, remember if we differentiate the answer with respect to x x then any function of only y y ’s will differentiate to zero.
What do you need to know about iterated integrals?
So, the iterated integral that we need to compute is, When setting these up make sure the limits match up to the differentials. Since the d y d y is the inner differential ( i.e. we are integrating with respect to y y first) the inner integral needs to have y y limits.