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📝 Double integral definition Riemann sums (13 MCQs)

📖 From Calculus • 15. Multiple Integrals Calculus • 13 questions available

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What is Double integral definition Riemann sums?

Definition:
A double integral is defined as the limit of Riemann sums: $\iint_R f(x, y) \, dA = \lim_{m,n \to \infty} \sum_{i=1}^m \sum_{j=1}^n f(x_{ij}^*, y_{ij}^*) \Delta A$ , where $\Delta A$ is the area of each subrectangle.

Example:
If we partition a square region into small squares of area $\Delta A = 0.01$ and sum the values of $f(x,y) = x^2 + y^2$ at sample points, the sum approaches the exact integral as the grid becomes finer.

Reason:
This foundational definition ensures that the integral represents the accumulated total of the function values over the region, providing rigorous mathematical justification for calculation methods.

Easy

Medium

Hard

📝 All Double integral definition Riemann sums MCQs

Q1. A function $f(x,y)$ is continuous on a rectangular region $R=[a,b]\times[c,d]$ . Which interpretation most accurately describes $\iint_R f(x,y)\,dA$ ?

A.The average value of

f

multiplied by the perimeter of

R

B.The signed accumulation of

f

over every infinitesimal area element of

R

✅

C.The maximum value of

f

multiplied by the area of

R

D.The sum of

f

only along the boundary of

R

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: A double integral accumulates the values of a function across a two-dimensional region. Each small area element contributes approximately $f(x,y)\Delta A$ , and the limiting process produces $\iint_R f(x,y)\,dA$ . Thus, it represents a signed accumulation over the entire region rather than only its boundary or extreme values.

Q2. For a rectangular region $R$ , a student writes $\iint_R f(x,y)\,dA=\lim\sum f(x_i,y_i)\Delta x\Delta y$ . What essential feature is missing from this definition?

A.The function must be replaced by its maximum value

B.The partition must become finer so the largest subrectangle area approaches zero ✅

C.The limits must always be circular

D.The function must be evaluated only at boundary points

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: A double integral is defined through a limit of double Riemann sums. The partition must become increasingly fine, meaning the dimensions and therefore the areas of the subrectangles approach zero. Without this limiting refinement, the expression is only a particular finite approximation rather than the double integral itself.

Q3. Suppose $f(x,y)$ represents the net vertical rate of material accumulation per unit area on a thin rectangular sheet. What does $\iint_R f(x,y)\,dA$ represent when positive and negative values are physically meaningful?

A.The total net accumulation over the sheet ✅

B.The largest local accumulation rate

C.The average accumulation rate at the center

D.The length of the boundary where accumulation occurs

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: When $f(x,y)$ represents a signed quantity per unit area, multiplying by a small area gives the corresponding local contribution. Summing those contributions and taking the limiting process gives the total net accumulation. Positive and negative values can therefore cancel, so the result is not necessarily the total absolute amount.

Q4. A rectangular region is divided into $400$ equal small rectangles. A student samples $f$ at one point in each rectangle and multiplies every sample by the common area before adding. What is this calculation best described as?

A.An exact double integral in every case

B.A finite approximation to the double integral ✅

C.A derivative of the double integral

D.A boundary integral

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The calculation is a double Riemann sum, which approximates the double integral. It becomes an exact representation only in the limit as the partition is refined appropriately. Using a finite number of rectangles generally introduces approximation error because the function can vary within each individual rectangle.

Q5. A temperature field is modeled by $T(x,y)=20+x^2-y$ on a metal plate. An engineer wants the total temperature-weighted area contribution over a specified region $R$ . Which mathematical model is most appropriate?

\int_{\partial R}T\,ds

\iint_R T(x,y)\,dA

✅

T(0,0)\operatorname{Area}(R)

\max_R T-\min_R T

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The quantity is distributed throughout the interior of the plate, so each small area contributes according to the local temperature. A double integral combines those local contributions across the entire region. A boundary integral would use only the edge, while evaluating at one point ignores variation across the plate.

Q6. A student argues that if $f(x,y)$ takes both positive and negative values on $R$ , then $\iint_R f(x,y)\,dA$ must equal the total amount of area between the graph of $f$ and the $xy$ -plane. What is the flaw?

A.Double integrals cannot involve negative values

B.Positive and negative contributions may cancel in the signed integral ✅

C.The graph must always be a plane

D.Area between surfaces is determined only by the boundary

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: A double integral of a signed function produces net accumulation. Regions where $f$ is positive contribute positively, while regions where $f$ is negative contribute negatively. Consequently, cancellation can occur. The total geometric area between the graph and the $xy$ -plane would instead require integrating the absolute value $|f|$ .

Q7. A function $f(x,y)$ is larger near the left side of a region and smaller near the right side. Two students use the same number of equal-area subrectangles. Student A samples near the left edge of each rectangle, while Student B samples near the right edge. Which statement is most accurate?

A.Both approximations must always be identical

B.They can differ because different sample points can produce different finite approximations ✅

C.Student A must always obtain the exact integral

D.Student B cannot form a Riemann sum

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: A finite double Riemann sum depends on the selected sample point in each subrectangle. If the function varies substantially, choosing points toward different sides can change the approximation. As the partition becomes sufficiently fine, these reasonable Riemann sums converge to the same double integral under the usual integrability conditions.

Q8. A graph of $f(x,y)$ over a rectangular region shows a smooth surface that rises steadily from one corner to the opposite corner. A student claims that the double integral is determined only by the highest point of the surface. Which conclusion is correct?

A.The highest point alone determines the integral

B.The lowest point alone determines the integral

C.The integral depends on the accumulated values across the entire region ✅

D.The integral equals the height difference between the two corners

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: A double integral incorporates contributions from every small area element in the region. A highest point provides only one function value and cannot determine the accumulated quantity by itself. Even for a smoothly varying surface, the integral depends on the function's behavior throughout the complete two-dimensional domain.

Q9. Consider two functions $f$ and $g$ on the same region $R$ . Suppose $f(x,y)\ge g(x,y)$ everywhere on $R$ , with strict inequality on a region having positive area. Which conclusion follows?

A.Their double integrals must be equal

B.The double integral of

f

is greater than that of

g

✅

C.The double integral of

g

must be positive

D.Only the boundary values determine which integral is larger

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Because $f$ is at least as large as $g$ at every point, each corresponding small-area contribution from $f$ is at least as large. Since the inequality is strict over a region with positive area, the accumulated difference is positive. Therefore, $\iint_R f\,dA>\iint_R g\,dA$ .

Q10. A rectangular region has area $12$ . The function $f(x,y)=5$ everywhere on the region. A student estimates the double integral using a fine partition and obtains a value close to $60$ . Why is this result consistent with the definition?

A.Every sample value is

5

, so the accumulated contribution is

5

times the total area ✅

B.The partition contains exactly

60

rectangles

C.The integral equals the perimeter multiplied by

5

D.The function must first be differentiated

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: For a constant function, every small area element contributes the same value multiplied by its area. Summing these contributions gives $5$ times the total area, which is $5(12)=60$ . Refining the partition does not change this result because the function has no spatial variation.

Q11. A student computes $\iint_R f(x,y)\,dA$ by adding $f(x_i,y_i)$ over all sample points but never multiplying by the corresponding subarea. Why is the reasoning incorrect?

A.Function values alone do not measure the contribution of each two-dimensional piece ✅

B.The sample points must lie outside

R

C.Double integrals require only boundary values

D.Multiplication by area is needed only for one-dimensional integrals

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: A function value describes an amount per unit area when interpreted through a two-dimensional accumulation model. Each small region therefore contributes approximately $f(x_i,y_i)\Delta A_i$ , not merely $f(x_i,y_i)$ . Omitting the area factor removes the geometric scaling required by the definition of a double integral.

Q12. Suppose a square region is partitioned into increasingly smaller equal squares. For each partition, the student chooses arbitrary sample points and forms a corresponding sum $S_n$ . What property should $S_n$ have if $f$ is continuous on the square?

A.It should approach a unique limiting value as the partition becomes finer ✅

B.It must become exactly zero after enough subdivisions

C.It must increase without bound

D.It must depend permanently on the first sample point chosen

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Continuity ensures that the function does not exhibit uncontrolled jumps on the compact rectangular region. As the subareas become smaller, the variation of the function within each subregion becomes correspondingly manageable. Consequently, reasonable Riemann sums converge toward the same unique limiting value, which defines the double integral.

Q13. A square region is divided into four equal parts. The function is positive on two parts and negative on the other two, with equal magnitudes and equal areas in corresponding parts. Without performing detailed calculations, what can be inferred about the double integral if the positive and negative contributions exactly balance?

A.It must be positive

B.It must be negative

C.It must be zero ✅

D.It must equal the total geometric area

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: A double integral represents signed accumulation. If the positive contributions from some parts exactly balance the negative contributions from other parts, their effects cancel in the sum. Therefore, even though the function is not zero throughout the region and may have substantial positive and negative values, the resulting double integral can be exactly zero.

🔗 Related Topics (MCQs)

📝 Double Integrals in Calculus MCQs 📝 Volume under surface double integral MCQs 📝 How to evaluate double integrals MCQs 📝 Fubini's theorem for double integrals MCQs 📝 Properties of Double Integrals MCQs

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