Homework Statement
Complete the proof by using the Fundamental Theorem of Calculus TWICE to establish
\int_c^d(\int_a^b f _{x}(x,y)dx)dy=...=\int_a^b(\int_{c}^{d}f_{x}(x,y) dy)dx
Homework Equations
I know that the FTC states that if g(x)=\int_a^x(f), then g'=f
The Attempt at a...
By the Fundamental Theorem of Calculus, if
I(a) = \int_0^a \frac{1}{\sqrt{1-x^2}}\ dx, then
I'(a)=1/(sqrt{1-x^2}.
Then, since 0<x<1, we see that sqrt(1-x^2)>0.
Thus, I is increasing since I'(a) is positive.
I'm still not sure how to show that I is bounded by 2...
Homework Statement
Suppose 0<a<1.
1) Show that
0<Integral(0 to a)1/(sqrt(1-x^2)<=Integral(0 to a)1/(sqrt(1-x)<=2
2) Show that I(a)=Integral(0 to a)1/(sqrt(1-x^2) is increasing and bounded by 2.
3) Deduce that Integral(0 to 1)1/(sqrt(1-x^2) exists and has an improper integral...