# Fundamental theorem of calculus

• Lee33
In summary, we have two closed intervals in ##\mathbb{R}##, a continuous real valued function ##f##, and two existing integrals. The fundamental theorem of calculus is used to find the derivative of the integral function and apply it to solve the problem at hand.
Lee33

## Homework Statement

Let ##[a,b]## and ##[c,d]## be closed intervals in ##\mathbb{R}## and let ##f## be a continuous real valued function on ##\{(x,y)\in E^2 : x\in[a,b], \ y\in[c,d]\}.## We have that ##\int^d_c\left(\int^b_af(x,y)dx\right)dy## and ##\int^b_a\left(\int^d_cf(x,y)dy\right)dx## exist.

None

## The Attempt at a Solution

I am wondering how this was solved?

Why is that from the fundamental theorem of calculus we get ##\frac{d}{dt}\int^t_a\left(\int^d_cf(x,y)dy\right)dx## ##=## ##\int^d_cf(t,y)dy## ?

Lee33 said:

## Homework Statement

Let ##[a,b]## and ##[c,d]## be closed intervals in ##\mathbb{R}## and let ##f## be a continuous real valued function on ##\{(x,y)\in E^2 : x\in[a,b], \ y\in[c,d]\}.## We have that ##\int^d_c\left(\int^b_af(x,y)dx\right)dy## and ##\int^b_a\left(\int^d_cf(x,y)dy\right)dx## exist.

None

## The Attempt at a Solution

I am wondering how this was solved?

Why is that from the fundamental theorem of calculus we get ##\frac{d}{dt}\int^t_a\left(\int^d_cf(x,y)dy\right)dx## ##=## ##\int^d_cf(t,y)dy## ?

Let $F: [a,b] \to \mathbb{R}: x \mapsto \int_c^d f(x,y)\,dy$, and apply the fundamental theorem of calculus to $\frac{d}{dt}\int_a^t F(x)\,dx$.

I am not sure what exactly to do. Can you elaborate further please?

Lee33 said:
I am not sure what exactly to do. Can you elaborate further please?

Can you please cite what exactly you mean with "the fundamental theorem of calculus"?

1 person

## What is the fundamental theorem of calculus?

The fundamental theorem of calculus is a fundamental concept in calculus that establishes the relationship between differentiation and integration. It states that the integral of a function can be evaluated by finding the antiderivative of the function and evaluating it at the upper and lower limits of integration.

## How is the fundamental theorem of calculus applied?

The fundamental theorem of calculus is applied in various fields of mathematics, physics, and engineering to solve problems involving rates of change and accumulation. It is also used to find the area under a curve and to evaluate definite integrals.

## What are the two parts of the fundamental theorem of calculus?

The fundamental theorem of calculus is divided into two parts: the first part, also known as the fundamental theorem of calculus, states that the derivative of the definite integral of a function is equal to the original function. The second part, also known as the fundamental theorem of calculus for line integrals, is used to evaluate line integrals.

## What is the difference between the fundamental theorem of calculus and the chain rule?

The fundamental theorem of calculus and the chain rule are two different concepts in calculus. The fundamental theorem of calculus relates to the relationship between differentiation and integration, while the chain rule is used to find the derivative of composite functions. However, the fundamental theorem of calculus can be used to prove the chain rule.

## What are some real-life applications of the fundamental theorem of calculus?

The fundamental theorem of calculus has various real-life applications, such as calculating the velocity of an object from its acceleration, finding the distance traveled by a moving object, and determining the rate of change of a population. It is also used in physics to calculate work, energy, and power.

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