Fundamental theorem of calculus

[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.

Homework Equations

None

The Attempt at a Solution

I am wondering how this was solved?

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

 

[pasmith]

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.

Homework Equations

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 $\frac{d}{dt}\int_a^t F(x)\,dx$.

 

[Lee33]

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

 

[micromass]

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"?