Improper Integrals [7.3]

$f(x)=1/x^2$

1. Sketch the function $f(x)=1/x^2$.
2. Graphically, What area is represented by the definite integral $$\int_1^{\infty}\frac 1{x^2}\,dx?$$ show / hide
3. Try to estimate the integral: Find the anti-derivative, and then calculate...
   $\int_1^{10}\frac 1{x^2}\,dx$
   $\int_1^{100}\frac 1{x^2}\,dx$
   ...
   $\int_1^{100,000}\frac 1{x^2}\,dx$

   Do you think there's a limit?

$f(x)=1/x$

1. Sketch the function $f(x)=1/x$.
2. Graphically, What area is represented by the definite integral $$\int_1^{\infty}\frac 1{x}\,dx?$$ show / hide
3. Try to estimate the integral: Find the anti-derivative, and then calculate...
   $\int_1^{10}\frac 1x\,dx$
   $\int_1^{100}\frac 1x\,dx$
   ...
   $\int_1^{100,000}\frac 1x\,dx$

   Do you think there's a limit?

Try these...

• $\int_0^{\infty}e^{-x}\,dx$
• $\int_4^{\infty}\frac{1}{\sqrt x}\,dx$