Concept explainers
Calculate the Riemann sum for the given
a) Lower-left vertex
b) Midpoint of rectangle
Then calculate the exact value of the double integral.

Answer to Problem 1CRE
Solution:
(a) The Riemann sum for the given double integral using lower-left vertices is 240.
(b)The Riemann sum for the given double integral using midpoints is 510.
And the exact value of the double integral is 520.
Explanation of Solution
Given:
The integral:
Formulas:
Where
Calculations:
From the given integral, we can observe that and . Since our aim is to find , we need to divide the rectangle into subrectangles. The length and width of each subrectangle are calculated as follows:
Therefore, the area of each subrectangle is .
The subrectangles are shown in Image 1.
Image 1:
(a) Using Lower-left vertex
Here, we use the lower-left vertices of each subrectangleto find the Riemann sum . Notice that the lower-left vertices are and are shown in Image 2.
Image 2:
Thus,
(b) Using Midpoint of Rectangle:
Here, we use the midpoints of each subrectangle to find the Riemann sum . Notice that the midpoints are and are shown in Image 3.
Image 3:
Thus,
To calculate the exact value of the integral:
Conclusion:
Thus,
(a) The Riemann sum for the given double integral using lower-left vertices is 240.
(b)The Riemann sum for the given double integral using midpoints is 510.
And the exact value of the double integral is 520.
Want to see more full solutions like this?
Chapter 16 Solutions
CALCULUS (CLOTH)
- Let f(x, y) = 2x + 3y+ In(xy)arrow_forward(3) (16 points) Let D = [0, π/2] × [0, 7/6]. Define T: DCR2 R3 by → T(0, 4) = (2 sin cos 0, 2 sin sin 0, 2 cos x). Let S be the surface parametrized by T. (a) (8 points) Determine the normal, call it n(p), for the tangent plane TS at an arbitrary point p = T(0, 4). (b) (4 points) Show that n(p) parallel to the position vector T(0, 4) determined by p? Do the two vectors have the same direction or opposite direction? Explain. (c) (4 points) At which points p, if any, is TS parallel to the xy-plane?arrow_forward5:19 0 TEMU TEMU >>> 49 95% University at Albany - Single Sig... L Lumen OHM D2L HW4- AMAT100-Precal HW4 Score: 12.99/21 Answered: 18/21 × Question 16 Score on last try: 0 of 1 pts. See Details for more. > Next question Get a similar question You can retry this question below Find the inverse for the function k(x) = √√7x+12 k-¹(x) = Question Help: Video Message instructor Submit Question esc ||| F1 80 ୮ (x) = tarrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage