Question

Evaluate the following improper integrals as iterated integrals.

(a) Z ∞

1

Z 1

e−x

1

x

3y

dy dx (b) Z 1

−1

Z √ 1

1−x2

√−1

1−x2

(2y + 1) dy dx

Answer 1

a)

$\intop^{\infin}_1\intop^1_{e^{-x}}\frac{1}{x^3y}dydx=\intop^{\infin}_1\intop^1_{e^{-1}}\frac{1}{x^3y}dydx=\intop^{\infin}_1\frac{lny}{x^3}|^1_{e^{-1}}dx=$

$=\intop^{\infin}_1\frac{1}{x^3}dx=-\frac{1}{2x^2}|^{\infin}_1=-1/2$

b)

$\intop^{1}_{-1}\intop^{1/\sqrt{1-x^2}}_{-1/\sqrt{1-x^2}}(2y+1)dydx=\intop^{1}_{-1}\intop^{\infin}_{-\infin}(2y+1)dydx=\intop^{1}_{-1}(y^2-y)|^{\infin}_{-\infin}dx=-\infin$