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.
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Chapter 16 Solutions
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