Let \( f(x, y) = x^2 y \) be a function defined over the square \( R \). Let the lower left corner be at \( (1, 0) \) and its upper right corner at \( (5, 4) \). Using a Riemann sum with \( m = 2 \), \( n = 4 \) as sample points in the lower right corners to estimate the value of \[\iint_R f(x, y) \, dA\]*sketch the region \( R \)*Explanation:- The function is \( f(x, y) = x^2 y \).- The region \( R \) is a rectangle with corners at \( (1, 0) \) and \( (5, 4) \).- The Riemann sum uses \( m = 2 \) subdivisions along the x-direction, and \( n = 4 \) subdivisions along the y-direction.- The sample points are the lower right corners of the resulting subrectangles.To compute the Riemann sum:1. Divide the width (x-direction) from 1 to 5 into 2 equal parts.2. Divide the height (y-direction) from 0 to 4 into 4 equal parts.3. Calculate the function \( f(x, y) \) at each sample point located at the lower right corner of each subrectangle.4. Sum the results to estimate the double integral over the region \( R \).*Sketch:The region \( R \) is a rectangle with vertices at:- \( (1, 0) \): Lower left corner- \( (5, 0) \): Lower right corner- \( (1, 4) \): Upper left corner- \( (5, 4) \): Upper right cornerInside the rectangle, there are subdivisions due to the Riemann sum criteria.

The sketch of the region R is defined as : The values are : ∆x=5-12=2∆y=4-04=1∆A=∆x∆y=2×1=2 The given function is : f(x,y)=x2yThe value of integral will be : ∬f(x,y)dA=∆Af(3,0)+f(3,1)+f(3,2)+f(3,3)+f(5,0)+f(5,1)+f(5,2)+f(5,3)=20+9+18+27+0+25+50+75=408 CONCLUSION: The value of integral in region R is 408.