Question

(a) Use a Riemann sum with m = n = 2 to estimate the value of the sin(x + y) dA R where R = [0, π] × [0, π]. Take the sample points to be lower left corners. (Round your answer to three decimal places.)

Answer 1

Answer:

4.9361

Step-by-step explanation:

Rieman sum = ∫∫$sin(x+y)$ΔA

ΔA = ΔxΔy

Δx = $\frac{π -0}{2}$  = $\frac{π}{2}$ = pi/2 = Δy

ΔA =  pi/2 * pi/2 = $\frac{pi^{2} }{4}$

Evaluating at the four corners

$f(0,0) + f(0, pi/2) + f(pi/2,0) + f(pi/2, pi/2)$ =

$sin(0 +0) + sin(0 + pi/2) + sin(pi/2 + 0) + sin(pi/2 , pi/2) = 0 +1 + 1 + 0$ = 2

∫∫$sin(x+y)$ΔA = 2 * $\frac{pi^{2} }{4}$  = $\frac{pi^{2} }{2}$ = 4.9361