Alan

Good point Guest! You are quite right. A better derivation is needed. 

In fact, I drew the graph first, and could see that the limit was zero, so my derivation of the limit was done rather hastily!

Here's a 3D graph to illustrate:

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The limit is zero.  See below:

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Distance in one turn of the wheel = circumference = pi*50 cm

Number of turns in 1km = 100000/(50pi).   Because there are 100000 cm in 1km

1 + 1.1p - (1.5p - 1) = 4

-0.4p + 2 = 4

-0.4p = 2

p = 2/(-0.4)

p = -5

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The dummy variable under the integral sign is t, but the integral itself is a function of x, and the derivative is taken with respect to x.

Imagine a simpler situation where you have d/dx of the integral from 0 to x of t with respect to t.

The integral would result in x^2/2, which, when differentiated wrt x would return x (not t).

Equate the two functions to get x^2 = m*x - 2   or  x^2 - m*x + 2 = 0

This quadratic has a single point solution if (-m)^2 - 4*1*2 = 0  or m^2 = 8  

Therefore there are two possibilities:  m = sqrt(8) and m = -sqrt(8)

These mean that the linear function represents tangent lines to the x^2 function.

It's the area of a semicircle of radius 3.  This is (1/2)*pi*3^2   or just 9pi/2.

Essentially correct User101, though the result should be expressed in terms of x rather than t.

Not all possible functions are implemented in the calculator.  You tend to get TO DO when you try a function that hasn't yet been implemented (and possibly, that Admin intends to implement at some time in the future?).