Why doesn't the derivative of an integral give the value at the lower limit?

[barksdalemc]

Can someone explain a concept to me? The derivative of an integral ( whose lower limit is a real constant and whose upper limit is the variable x), is the intergrand evaluated at x as per the FTofC. I always thought about this as the limit of the integral as x approached the lower limit becuase by definition of the derivative we take limit as change in x approaches 0. So my question is why the derivate of an integral doesn't give the value of the function at the lower limit.

 

[matt grime]

Why did you think this? It is not correct. The limit of
$$\int_c^x f(t)dt$$

as x tends to c is zero.

What you shuld be thinkig about is

$$\frac{1}{h} (\int_c^{x+h} f(t)dt - \int_c^x f(t)dt)$$

as h tends to zero.

 

[barksdalemc]

Yes that makes sense. So becuase the lower limit is fixed the rate of change of the area under the curve to the lower limit is zero, but the rate of change of the area up to the upper limit is changing by a value equal the integrand value evaluated at the upper limit?

 

[barksdalemc]

I just saw the second equation you posted. That makes it 100% clear to me. Thanks.