Double Integrals over Rectangular Regions

1. Calculate the iterated integral.

 (a)  \(\displaystyle{ \int_1^4 \int_0^2 (6x^2y-2x)y \,dy\, dx} \)    solution

(b)  \(\displaystyle{ \int_0^{\pi/6} \int_0^{\pi/2} (\sin x + \sin y) \,dy\, dx} \)   solution

(c)  \(\displaystyle{ \int_1^{4} \int_1^{5} \frac{\ln y}{xy} \,dy\, dx} \)    solution

(d)  \(\displaystyle{ \int_1^e \int_1^{2} x^2 \ln x \,dy\, dx} \)   solution

2. Evaluate \(\displaystyle{ \iint_R x e^{xy}\, dA} \), where  \(\displaystyle{R=[0, 1]\times[0,1]} \).  solution

3. Find the average value of the function \( \displaystyle{f(x,y)=x^4+2y^3}\) over the  rectangle \(\displaystyle{R=[1, 2]\times[2,3]} \).  solution

4. Compute \(\displaystyle \iint_R \cos(x+y)\,dA\), where \(R\) is the rectangle in the \(xy\)-plane determined by the inequalities \(0\le x\le \frac{\pi}{6}\) and \(0 \le y \le \frac{\pi}{2}\).  solution