Exercise · ZW-DOUBLE-INTEGRAL-004 · ★★☆ Intermediate
With the same partition of \(R=[0,2]\times[0,2]\) into four squares, compute the Riemann sum of \(f(x,y)=x+y\) taking the center of each square.
Choose one option.
Show answer
- A. \(8\)
Explanation
The centers are \((\tfrac12,\tfrac12)\), \((\tfrac32,\tfrac12)\), \((\tfrac12,\tfrac32)\), \((\tfrac32,\tfrac32)\), with values \(1,2,2,3\) (sum \(8\)) and \(\Delta A=1\). The result is \(8\), which equals the exact value of the integral because \(f\) is linear.
Review the concept: Double integral
The double integral of a function \(f\) over a region \(D\) of the plane is the limit of the sums of the values of \(f\) times the area \(\Delta A\) of ever finer cells covering \(D\).
\[\iint_D f(x,y)\,dA=\lim_{\max\Delta A_i\to0}\sum_i f(x_i^*,y_i^*)\,\Delta A_i\]
More practice problems: double integral
- ZW-DOUBLE-INTEGRAL-001 Compute iint_R 5 dA over the rectangle R=[1,4]×[0,2].
- ZW-DOUBLE-INTEGRAL-002 Find the volume of the solid under the plane z=4 and above the disk D of radius 2 centered at the o…
- ZW-DOUBLE-INTEGRAL-003 Divide R=[0,2]×[0,2] into four squares of side 1 and compute the Riemann sum of f(x,y)=x+y taking t…
- ZW-DOUBLE-INTEGRAL-005 If iint_D f dA=3 and iint_D g dA=5, what is iint_D (2f-g) dA?
- ZW-DOUBLE-INTEGRAL-006 The region D is the union of two non-overlapping regions D_1 and D_2. If iint_D_1 f dA=7 and iint_D…
- ZW-DOUBLE-INTEGRAL-007 On a region D of area 6 it is known that iint_D f dA=24. What is the average value of f on D?
- ZW-DOUBLE-INTEGRAL-008 Compute iint_D x dA over the unit disk D:\ x^2+y^2≤1.
- ZW-DOUBLE-INTEGRAL-009 If 1≤ f(x,y)≤3 on a region D of area 5, which inequality holds for iint_D f dA?