Exercises

01

Reversing integration order

Check that Fubini’s Theorem holds: evaluate ${\int }_0^3{\int }_1^2{x}^{2}y+x{e}^{y}dydx$.

02

Iterated integral practice

Evaluate the iterated integrals:

• (a) ${\int }_0^{\pi }{\int }_0^{\sin (x)}(1+\cos (x))dydx$
• (b) ${\int }_0^2{\int }_{3y^2-6y}^{2y-y^2}3ydxdy$
• (c) ${\int }_0^{\pi /4}{\int }_{\sqrt{3}}^{\sqrt{3}\cos (\theta )}rdrd\theta$
• (d) Extending to 3D: ${\int }_0^1{\int }_0^y{\int }_x^{1}12xyzdzdxdy$

03

Volume of a solid

Find the volume of the region under $f(x,y)=x^{2}y$ for $x\in [1,3]$ and $y\in [x,2x+1]$.

04

Change order of integration

Change the order of integration in the following double integral:

$${\int }_1^9{\int }_{\sqrt{y}}^{3}x{e}^{y}dxdy.$$

(Hint: first sketch the region in question.)