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

• barksdalemc
In summary, the derivative of an integral with a fixed lower limit and variable upper limit is equal to the integrand evaluated at the upper limit. This is because the limit of the integral as the upper limit approaches the lower limit is zero, while the rate of change of the area under the curve up to the upper limit is equal to the integrand value at the upper 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.

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.

Last edited:
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?

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

## What is the Fundamental Theorem Question?

The Fundamental Theorem Question is a mathematical concept that relates to the fundamental theorem of calculus. It states that if a function is continuous on a closed interval, then the definite integral of that function over that interval is equal to the difference between the antiderivative of the function evaluated at the upper and lower limits of the interval.

## Why is the Fundamental Theorem Question important?

The Fundamental Theorem Question is important because it provides a way to evaluate definite integrals without having to use complicated methods, such as the limit definition of a definite integral. It also allows for the calculation of areas and volumes in real-world applications.

## What is the difference between the Fundamental Theorem Question and the Fundamental Theorem of Calculus?

The Fundamental Theorem Question is a specific question that relates to the fundamental theorem of calculus, which is a broad mathematical concept. The fundamental theorem of calculus states that the derivative of an antiderivative of a function is equal to the original function.

## How is the Fundamental Theorem Question used in real-world applications?

The Fundamental Theorem Question is used in various fields such as physics, engineering, and economics to calculate areas, volumes, and other quantities that involve integration. It is also used in optimization problems to find the maximum or minimum value of a function.

## Can the Fundamental Theorem Question be applied to all functions?

The Fundamental Theorem Question can be applied to all continuous functions on a closed interval. However, it may not be applicable for some discontinuous or undefined functions. In those cases, other methods of integration may need to be used.

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